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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . | so.. what would 6 ( g+3 ) be ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? | what is 4 ( 6-8*9+5 ) +-5 ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? | i cut off a piece of length 4 '' what is the area of the remaining piece ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . | when will you ever need to use expressions in life ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . | what is the difference between the law of multiplication and the addition rule ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . | so we only use the distribution property when we have things in brackets ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . | when do i know to keep the brackets and when not too ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . | when do we decide to put/or have to put parenthesis and when not to ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . | what is and how do you do distributive property of division ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . | what is a distributed property/ law ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . | so , how would you answer a problem say its like 3 loaves of bread for $ 1.99 how would you awnser that with the distributive property ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . | does sal use the distributive law 7 ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . | so what exactly does distribute mean ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . | how would the distributive property work if the problem is ( 3-6 ) ^2 ( where the two is to the second power of ( 3-6 ) ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . | what is the commutative property ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . | can any one tell me how to apply the distributive property to everyday life or who uses this type of math ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? | why did sal say to add all the blue and red circles up when you could multiply them ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . | how do you use the distributive property using variables and numbers ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . | why is the distributive property so useful ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . | if you had an negative number would the negative be distributed to the equation inside the bracket or not ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . | also if you had two numbers in front of the bracket and the first was a positive and the second was an negative would both numbers be distributed or not ? |
rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition ... | rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . | does butt chicken tastes good ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . | what is the difference between a `` estimated '' number and a `` projected '' number ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major... | i 'm just thinking how bad this will be in 30 years from now when i retire , will the state cut/will i lose my promised pension altogether or , as selfish as it sounds , will the state still pay up at the cost of cutting major funding for other things like they have been doing ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road... | do n't you think a numbers guy should shop around for a better rate ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % intere... | does illinois report its funding ratio assuming the more reasonable rates of return expressed in the previous video and if so , what is its funded status at those rates of return ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . | why should the state fund pensions ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . | is n't it the company 's fault for having underfunded pensions ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pret... | can illinois get out of dept with out raising taxes ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . | when the inflation rate start to increase past 3 percent , would the cola automatically increase to match the new higher inflation rate ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . | my question is , are any of the state pension funds managed by our `` too big to fail '' brokerage firms , like j.p. morgan chase ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . | if we forced them all to take bailout money , why ca n't we ask them to responsibly manage pensions ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . | does this mean defined benefit pension plans will become dramatically underfunded all over again - at all levels of government including federal ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? | what is ma 's under funding percentage ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially ju... | we are always in a low rate but why ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . | why does your discussion assume that the only way a state can meet its pension obligations is by giving money to the private sector to invest ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % intere... | what would be wrong with paying-as-you-go : that is , paying each year 's obligations to retirees in the years in which they are entitled to pensions , instead of entering the weird world of predictions of ( a ) future benefit starting amounts , ( b ) future inflations rates , ( c ) future rates of investment return ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | what is the recommended level of funding by defined benefit plans ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . | also does n't the funding level get tricky with the return on investments being harder to project ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . | was the illinois pension fund invested in less safe investments than most pension funds and thus lost a lot of money during the 2008 financial crisis ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? | what is massachusetts u.f ( under funding ) percentage ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go int... | how much of the 2010 pension shortfall is artificially low because it is reflecting the low market value of stocks , funds , and other investments in 2010 because of the financial crisis in 2008 ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . | is the situation in ct similar to that described for illinois ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . | why is puerto rico not included in the map ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . | what are the benifits to not paying your prompised liabilities ? |
in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so rig... | this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . | why do governments utilize a different accounting system than private businesses ? |
let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ? all right , this is interesting . so , one way i could think about doing this is , well let 's just try each of these f of xes . if this is , if this was f ... | let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ? | what is a domain and range ? |
let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ? all right , this is interesting . so , one way i could think about doing this is , well let 's just try each of these f of xes . if this is , if this was f ... | let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ? | hi , my question is , how is x equal to g ( x ) ? |
let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ? all right , this is interesting . so , one way i could think about doing this is , well let 's just try each of these f of xes . if this is , if this was f ... | let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ? | i understand how you solve a function using operations , but how do you interpret the graph of a function ? |
let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ? all right , this is interesting . so , one way i could think about doing this is , well let 's just try each of these f of xes . if this is , if this was f ... | let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ? | how do you know what the value of a function is by looking at its graph ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | was the intuition of the mathematicians who defined matrix addition really that arbitrary ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way . | can matrices be considered as vectors , in a way , with the addition and subtraction properties of matrices being similar to that of vectors ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . | what are matrices used for ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way . | what is the point of a matrix ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , ... | could there ever be a situation when you have to add or subtract matrices of different dimensions ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . | what are the point of matrixes ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . | i understand why you would n't be able to put the matrices in any order while dividing , but since multiplying is simply repeated addition , would n't the order of two matrices not matter ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . | can we divide a matrice with one other ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is ... | if i want to add two matrices with dimensions 3 x 2 and 2 x 3 , can i transpose one of them to get two matrices with dimensions 3 x 2 and 3 x 2 and then perform addition ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , ... | or is it that such kinds of matrices with different dimensions can never be added ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scena... | so if adding and subtracting different dimensional matrices is undefined , can you multiply and divide ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . | why ca n't you just put a zero in all the missing places and then add all the others ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . | can we use properties such as the associative property or the distributive property while working with matrices ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . | how do find the additive inverse of matrices ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , ... | to add or subtract matrices of different dimensions , why would n't it work to fill in zeros in the matrices ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . | why is there a `` d '' after the equals sign ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . | is it just khan making a smiley face or is it used to indicate the sat choice `` d. undefined '' ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . | why ca n't we add zero as a place-holder for any missing rows/columns ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way . | how do you know if a matrix is unsolvable ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm... | does the size of a and b have to be the same in order to do these operations ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . | for instance , could n't we rewrite the 2x2 matrix [ 5 7 ; -1 6 ] as a 3x2 [ 5 7 ; -1 6 ; 0 0 ] ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logical... | if the addition and subtraction of different dimension matrices is undefined , is the same true for multiplication ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . | is it possible to multiply and divide matrices ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , ... | when dealing with arrays of different lengths ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way . | is it possible to divide or square root a matrix ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . | can there be a third variable of some sort in a matrix besides rows and columns ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . | how do you enter matrices into the computer ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . | do two matrixes have to have the same dimensions to be added ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) | waht would be the sum of 3+ [ a matrix ] ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way . | how do you do scalar matrix addition and subtraction ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , ... | why is it then not possible to add two matrices with different dimensions in this aggregate way ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . | how do you use matrices to solve nonmatrix problems in real life ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way . | for matrices with different dimensions , for instance a 2x3 matrix and 2x2 matrix , why ca n't we consider the latter to have a third column with all entries equal to zero so that we can add the two matrices ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . | would you change pi to 3.14 and add ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , ze... | what if i wanted to add a 3x4 matrix to a 12x1 matrix ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . | would [ 6 7 8 ] - [ 5 ] be undefined ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . | while multiplying two numbers with both negative signs , will it turn positive in the product part or negative ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . | is the final result of adding/subracting matrices always going to have only 1 column ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | what does the word `` arbitrary '' mean ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , ... | so you ca n't add matrices of different dimensions ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way . | is it possible for a matrix to have a complex number in it ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . | at the last part of the video , could n't you just expand one of the matrices by writing in 0 's ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way . | could n't the dimensions be added or subtracted the same way as the numbers within the matrix ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way . | are there any operations that can reduce or increase the initial size of an matrix ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way . | is matrix addition is possible when there is the number of rows or columns are different ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way . | how do you multiply a number by a matrix ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | what is an arbitrary number ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way . | i still ca n't figure out matrices where a matrix + b matrix = c matrix and it 's asking me to solve for x and y ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . | how do you solve matrices with variables ? |
let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices l... | and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to th... | the identity matrix can serve as a scalar , so why ca n't composed matrices be converted to bigger sizes ? |
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