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so we have a quadratic expression here . x squared minus three x minus 10 . and what i 'd like to do in this video is i 'd like to factor it as the product of two binomials . or to put it another way , i want to write it as the product x plus a , that 's one binomial , times x plus b , where we need to figure out what... | so plus bx . b x . and then finally we have plus the a times the b , which is of course going to be ab . | example : x^2-5x-14= ( x-7 ) ( x+2 ) am i missing something ? |
so we have a quadratic expression here . x squared minus three x minus 10 . and what i 'd like to do in this video is i 'd like to factor it as the product of two binomials . or to put it another way , i want to write it as the product x plus a , that 's one binomial , times x plus b , where we need to figure out what... | we have something times x , in this case it 's a negative three times x . and here we have something times x . so one way to think about it is that a plus b needs to be equal to negative three . | so what would the factors be for something like 4x squared +4x-3 ? |
so we have a quadratic expression here . x squared minus three x minus 10 . and what i 'd like to do in this video is i 'd like to factor it as the product of two binomials . or to put it another way , i want to write it as the product x plus a , that 's one binomial , times x plus b , where we need to figure out what... | so plus bx . b x . and then finally we have plus the a times the b , which is of course going to be ab . | what would you do if the expression is 3 ( x+y ) + a ( x+y ) ? |
so we have a quadratic expression here . x squared minus three x minus 10 . and what i 'd like to do in this video is i 'd like to factor it as the product of two binomials . or to put it another way , i want to write it as the product x plus a , that 's one binomial , times x plus b , where we need to figure out what... | so plus bx . b x . and then finally we have plus the a times the b , which is of course going to be ab . | how to factorize x^3 - x ? |
so we have a quadratic expression here . x squared minus three x minus 10 . and what i 'd like to do in this video is i 'd like to factor it as the product of two binomials . or to put it another way , i want to write it as the product x plus a , that 's one binomial , times x plus b , where we need to figure out what... | so plus bx . b x . and then finally we have plus the a times the b , which is of course going to be ab . | how to prove 2x^4 + x^3-14x^2-19x-6 is exactly divisible by x^2+3x+2 without actual division ? |
so we have a quadratic expression here . x squared minus three x minus 10 . and what i 'd like to do in this video is i 'd like to factor it as the product of two binomials . or to put it another way , i want to write it as the product x plus a , that 's one binomial , times x plus b , where we need to figure out what... | but if you were to multiply what we have on the right-hand side out it would be equal to , you 're going to have the x times the x which is going to be x squared . then you 're going to have the a times the x , which is ax . and then you 're going to have the b times the x , which is bx . | is it possible to factor a polynomial ax^3+0x^2+cx+d ? |
so we have a quadratic expression here . x squared minus three x minus 10 . and what i 'd like to do in this video is i 'd like to factor it as the product of two binomials . or to put it another way , i want to write it as the product x plus a , that 's one binomial , times x plus b , where we need to figure out what... | so we have a times b needs to be equal to negative 10 . and in general , whenever you 're factoring something , a quadratic expression that has a one on second degree term , so it has a one coefficient on the x squared , you do n't even see it but it 's implicitly there . you could write this as one x squared . a way t... | what if the coefficient of the second degree term ( x squared ) is a natural number other than one ? |
so we have a quadratic expression here . x squared minus three x minus 10 . and what i 'd like to do in this video is i 'd like to factor it as the product of two binomials . or to put it another way , i want to write it as the product x plus a , that 's one binomial , times x plus b , where we need to figure out what... | so one way to think about it is that a plus b needs to be equal to negative three . they need to add up to be this coefficient . so let me write that down . | so will the product be that coefficient into the the constant ? |
so we have a quadratic expression here . x squared minus three x minus 10 . and what i 'd like to do in this video is i 'd like to factor it as the product of two binomials . or to put it another way , i want to write it as the product x plus a , that 's one binomial , times x plus b , where we need to figure out what... | so plus bx . b x . and then finally we have plus the a times the b , which is of course going to be ab . | is there a good way to remember that the constant needs to be the product of a and b and that the coefficient on the middle term needs to be the sum of a and b ? |
so we have a quadratic expression here . x squared minus three x minus 10 . and what i 'd like to do in this video is i 'd like to factor it as the product of two binomials . or to put it another way , i want to write it as the product x plus a , that 's one binomial , times x plus b , where we need to figure out what... | plus ab . and now we can use this to think about what a and b need to be . if we do a little bit of pattern matching , we see we have an x squared there , we have an x squared there . | how do you use factoring for a rational function ? |
so we have a quadratic expression here . x squared minus three x minus 10 . and what i 'd like to do in this video is i 'd like to factor it as the product of two binomials . or to put it another way , i want to write it as the product x plus a , that 's one binomial , times x plus b , where we need to figure out what... | so two numbers that add up to negative three , to add up to the coefficient here . and now when i multiply it , i get the constant term . i get this right over here . | why do i sometimes get an answer that is not the sum of the middle term but is the product of the last term ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | dr. steven zucker : right . but also these very complex leaf-like forms , which you can just make out here , which is actually from the acanthus leaf . and we have a photograph of an acanthus leaf right down there . dr. beth harris : and these grow wild so it makes sense . | is there a myth , or other explanation , on why the greeks chose the acanthus leaf , as opposed to , say , a grape leaf , for the corinthian capitals ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | and there you have it . the greek orders . | are there examples of buildings that combine different aspects of the orders together , and if not , then why are there these sudden shifts in architecture to such specific styles ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | dr. beth harris : let 's start at the top , with the pediment . the pediment is n't , officially , part of the order . but since greek temples had , at one end or the other , a pediment , we just thought we would name that for you . | why is n't there a `` phrygic '' order ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | dr. beth harris : it 's really fascinating to think about all of these decisions that the greeks are making as they build . so let 's look at the ionic order , which emerges shortly after the doric order . here 's another building of the acropolis , this is the erechtheion . | did the greeks cease to use the doric order when the ionic and corinthian came along ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | and there you have it . the greek orders . | is there a reason why doric ( even being the oldest of the orders ) seems to be the sturdiest ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | but they are highly decorative . there 's a great myth about the origin of the corinthian capital . dr. beth harris : it 's a kind of fun story . | how was the myth about the corinthian columns discovered ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . | i understand that the etruscan architecture was adapted from that of the greeks , but was the opposite true as well ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | dr. beth harris : and in fact , vitruvius the ancient roman architectural historian , saw this as a more feminine order -- it 's taller , it 's thinner . dr. steven zucker : now , one of the columns from this building in greece is in the museum in london . we have some good photographs of it . | were there any buildings in classical greece that implemented tuscan columns or any other elements of etruscan building styles ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . | does it seem reasonable to suggest that doric fluting lends to the visual illusion of height and as well as making the columns appear more parallel and less tapered ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | dr. steven zucker : and you know what ? we still use this basic system when we nail two-by-fours together . and that 's what the greeks were doing . but they were doing in a much more sophisticated way . | what building material did the greeks use ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | that is , the sides of a column are parallel with each other , and the base of a column is just as wide as the area directly below the capitol . but in fact , the ancient greeks did n't build their temples that way . dr. beth harris : no . | any specific reason why , or did the greeks just build round ones or rectangular ones willy-nilly ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | but we 've listed them here for you , just so you know what they are , the tuscan and the composite . so the doric and ionic and corinthian are illustrated , here , in this diagram . first the doric , and the ionic , and then , the last two are corinthian . | what is the really big difference between the doric , ionic , and the corinthian ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | dr. beth harris : the next area , below the pediment , is actually , officially part of the order . and that 's called the entablature . dr. steven zucker : ok , so that would be the area from about here to here . | so the top half of the entablature is the frieze , but what is the lower half of the entablature called or used for ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | because they get narrower as they go further up , it seems as if the shaft of the column might actually be taller than it really is . because of course , as things move away from us , they get smaller in scale . dr. beth harris : so the greeks are thinking about human perception . | why would the greeks want to make the columns appear smaller if the farther away we get the smaller it looks , as was mentioned ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's especially true of the classical orders . because these are what are , essentially , the building blocks of western architecture . | are the classical orders used in modern architecture ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | and there you have it . the greek orders . | is there any connection between the greek orders and arabian art , or was there no influence from either culture ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | and then , the corinthian , completely out of control . dr. beth harris : so let 's start with the oldest order , the doric order . dr. steven zucker : right , and we think that this order began in the seventh century , on the mainland in greece . | is the doric order the only place we see entasis , or do other orders use tapering columns as well ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | dr. beth harris : right . they developed decorative systems . and that 's what we 're referring to when we use the term classical orders . | why did the people make it more decorative than the other ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | but we 've listed them here for you , just so you know what they are , the tuscan and the composite . so the doric and ionic and corinthian are illustrated , here , in this diagram . first the doric , and the ionic , and then , the last two are corinthian . | is the fact that the corinthian shaft can be cuboid in shape ( and the fact that doric and ionic shafts are both circular ) important ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | first the doric , and the ionic , and then , the last two are corinthian . these are just slight variations of these three orders . dr. steven zucker : and the doric is really the most simple . | or can all three orders have cuboid shafts ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | and then , the corinthian , completely out of control . dr. beth harris : so let 's start with the oldest order , the doric order . dr. steven zucker : right , and we think that this order began in the seventh century , on the mainland in greece . and we 're looking at an actual greek temple that happens to be in italy... | 1 it 's mentioned that the doric order began in the seventh century on the greek mainland , when and where were the ionic and corinthian styles developed ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | dr. steven zucker : now the triglyphs we do n't think are just arbitrary . we think that they probably came from a time when temples were built out of wood . and these would have been the ends of planks that would have functioned as beams in the temple . | who built those temples anyway ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | but we 've listed them here for you , just so you know what they are , the tuscan and the composite . so the doric and ionic and corinthian are illustrated , here , in this diagram . first the doric , and the ionic , and then , the last two are corinthian . | why are the pillars called doric , ionic and corinthian ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | it 's fascinating to think about all the ways that the ancient greeks are thinking about how to make their buildings beautiful , and speak of the realm of the gods . and so , when we look at an ancient doric temple , we see that the shafts swell a little bit toward the center . dr. steven zucker : so right about a thir... | why do the shafts swell i am just wondering why there not straight ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | dr. beth harris : so you just used the word drum . so the columns are not , actually , carved from one piece of stone . dr. steven zucker : and if you look very carefully at this photograph , you can just make out the seams between those drums . | does the stone hedge inspire the buildings or temples ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | but i think it 's easy to not realize just how big they . but i snapped this terrific picture of you at the british museum next to a capital that actually comes from the most famous doric temple , on the acropolis in athens . dr. beth harris : right , the parthenon . | so acropolis , athens is a place ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | dr. beth harris : right . they developed decorative systems . and that 's what we 're referring to when we use the term classical orders . | why did the people make it more decorative than the other ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | dr. beth harris : and these grow wild so it makes sense . dr. steven zucker : what 's important to remember is that the ancient greeks , although they developed these three classical orders , were just the genesis . the romans took these ideas over . | how did the greeks built the classical orders ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | dr. beth harris : let 's start at the top , with the pediment . the pediment is n't , officially , part of the order . but since greek temples had , at one end or the other , a pediment , we just thought we would name that for you . | is n't there a connection between the classical order to ancient egyptian columnar structuring ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | and that 's what we 're referring to when we use the term classical orders . there are three basic orders , the doric , the ionic , and the corinthian . there 's a couple extra , but we 're not going to go into those today . | looking ; is it fair to say there is less or even no `` entasis '' or tapering in the ionic and corinthian orders ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | and it would taper , ever so slightly , towards the bottom , and taper much more so as we move up the top . so that the narrowest point of the column shaft would be right at the top . and the widest part would be about one third of the way from the base . | are the shafts diameter divided according to the golden ratio where the widest point divides the column ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | but we 've listed them here for you , just so you know what they are , the tuscan and the composite . so the doric and ionic and corinthian are illustrated , here , in this diagram . first the doric , and the ionic , and then , the last two are corinthian . | altogether how many feet/meters tall were the average doric , ionic and corinthian column ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | and there you have it . the greek orders . | was this one of the first buildings to juxtapose these styles , or were there examples during greek times too ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | dr. beth harris : so as we move down the temple , the next area we come to is the capital . dr. steven zucker : and this is a doric capital . it 's very simple . | in the trascription is `` capital '' or `` capitol '' ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | and there you have it . the greek orders . | why was greek people the smart ones ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | dr. beth harris : so let 's start with the oldest order , the doric order . dr. steven zucker : right , and we think that this order began in the seventh century , on the mainland in greece . and we 're looking at an actual greek temple that happens to be in italy . | why did roman take over greece ? |
dr. steven zucker : architecture is a language . and you know how when you learn a new vocabulary word , you start to notice it , for the first time , everywhere ? well , the same thing happens with architecture . when you learn a new architectural form , you start to see it everywhere . dr. beth harris : and that 's e... | and there you have it . the greek orders . | and finally why did the greek gods get their names changed ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | 6 plus 2 times minus 3 , that 's 6 minus 6 , that 's just 0 . and there we 've actually put our matrix in reduced row echelon form . so let me put brackets around it . | so every matrix in reduced row echelon form is linearly independent ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | when you put it in reduced row echelon form , it 's very clear that any pivot column will be the only one to have 1 in that place . so it 's very clear that these guys are linearly independent . now it turns out , and i have n't proven it to you , that the corresponding columns in a -- this is r1 , but it 's a before w... | is n't it clear that the vectors are linearly dependent ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | fair enough . now we want to zero out this guy . well it seems like a pretty straightforward way . | does the rank of a also equal to number of non-zero rows of rref ( a ) ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | and i 'll do a better explanation of this , but i really just wanted you to understand how to develop a basis for the column space . so they 're linearly independent . so the next question is do they span our column space ? | can the column space ever not be linearly independent ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | remember the basis just means that vectors span , c , a . clearly these vectors span our column space . i mean the span of these vectors is the column space . | if yes , if the column space vectors are not linearly independent , meaning there is not a zero above and below every pivot , what happens ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | if we just took a set of , let 's call this r1 , r2 , and this would be r3 , this would be r4 right here . it 's clear that the set r1 , r2 , and r4 is linearly independent . and you say why is that ? | are those columns ( r1 , r2 , and r4 ) dependent variables though as they have a leading 1 value ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | 6 plus 2 times minus 3 , that 's 6 minus 6 , that 's just 0 . and there we 've actually put our matrix in reduced row echelon form . so let me put brackets around it . | do you really need to get the matrix to reduced row echelon form to find the basis , i mean ca n't you just get the matrix in row echelon form and find the basis from there ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | remember the basis just means that vectors span , c , a . clearly these vectors span our column space . i mean the span of these vectors is the column space . | and i even can not define the column space ... also , q2 ) why i cant say that the pivot columns from n ( rref ( a ) ) span n ( a ) ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | this is by definition of a pivot entry . when you put it in reduced row echelon form , it 's very clear that any pivot column will be the only one to have 1 in that place . so it 's very clear that these guys are linearly independent . | simply put , how is one supposed to place vectors in a matrix , by column or by row ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | but for this case , if you want to know the basis , it 's just a1 , a2 , and a4 . and now we can answer another question . so a1 , a2 , and a4 form a basis for the column space of a , because you can construct the other two guys with linear combinations of our basis vectors , and they 're also linearly independent . | my question is , why take that extra step at the end ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | let me square them out , or circle them out . column 1 is a pivot column , column 2 is a pivot column , and column 3 is a pivot column . and we 've done this in previous videos . | i still do n't get how do you term a pivot column ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | minus 1 minus 2 times 1 is minus 3 . all right . now this last guy we want to eliminate him , and we want turn this into a 0 . | so the marix is linearly dependent and therefore it cant be a basis , right ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | now what do we see about matrix r ? well it has 3 pivot entries , or 3 pivot columns . let me square them out , or circle them out . | are the pivot columns of rref ( a ) also a basis of c ( a ) ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | remember the basis just means that vectors span , c , a . clearly these vectors span our column space . i mean the span of these vectors is the column space . | and i even can not define the column space ... also , q2 ) why i cant say that the pivot columns from n ( rref ( a ) ) span n ( a ) ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | remember the basis just means that vectors span , c , a . clearly these vectors span our column space . i mean the span of these vectors is the column space . | and i even can not define the column space ... also , q2 ) why i cant say that the pivot columns from n ( rref ( a ) ) span n ( a ) ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | remember the basis just means that vectors span , c , a . clearly these vectors span our column space . i mean the span of these vectors is the column space . | and i even can not define the column space ... also , q2 ) why i cant say that the pivot columns from n ( rref ( a ) ) span n ( a ) ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | remember the basis just means that vectors span , c , a . clearly these vectors span our column space . i mean the span of these vectors is the column space . | and i even can not define the column space ... also , q2 ) why i cant say that the pivot columns from n ( rref ( a ) ) span n ( a ) ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | so if i write it like this , a1 , a2 , and a4 . let me write it in set notation . these guys are also linearly independant , which i have n't proven . | give the an example for the orthogonal set and orthonormal set ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | and another way to think about it is , the rank of a is the number of linearly independent column vectors that you have that can span your entire column space . or the number of linearly independent column vectors that can be used to construct all of the other column vectors . but hopefully this did n't confuse you too... | is a linearly independent set the minimum amount of column vectors needed to describe a particular subspace ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | because any linear combination of them , or linear combinations of them can be used to construct the non-pivot columns , and they 're linearly independant . so i have n't shown you that . but for this case , if you want to know the basis , it 's just a1 , a2 , and a4 . | i want to know if im understanding it the way i should : is cardinality the minimum number of equations ( of n components ) required to describe an r^n sub-space/null-set ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | let me square them out , or circle them out . column 1 is a pivot column , column 2 is a pivot column , and column 3 is a pivot column . and we 've done this in previous videos . | is a column vector really a vector ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | remember the basis just means that vectors span , c , a . clearly these vectors span our column space . i mean the span of these vectors is the column space . | what 's the difference between row vectors and column vectors ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | now it turns out , and i have n't proven it to you , that the corresponding columns in a -- this is r1 , but it 's a before we put it in reduced row echelon form -- that these guys right here , so a1 , a2 , and a4 are also linearly independent . so a1 -- let me circle it -- a2 , and a4 . so if i write it like this , a1... | why are a1.a2 and a4 li ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | and the dimension of a column space actually has a specific term for it , and that 's called the rank . so the rank of a , which is the exact same thing as the dimension of the column space , it is equal to 3 . and another way to think about it is , the rank of a is the number of linearly independent column vectors tha... | is rank and range the same thing ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | well it seems like a pretty straightforward way . just replace this row with this row plus the first row . so minus 1 plus 1 is 0 . | is there any additional information concerning the row space of a matrix ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | now what do we see about matrix r ? well it has 3 pivot entries , or 3 pivot columns . let me square them out , or circle them out . | and for c ( a ) the number of bases equals to the numbers of the columns of the pivot entries ? |
we 've seen in several videos that the column space of a matrix is pretty straightforward to find . in this situation the column space of a is just equal to all of the linear combinations of the column vectors of a . another way of saying all of the linear combinations is just the span of each of these column vectors .... | so we have 1 , 2 , 3 vectors . so the dimension of our column space is equal to 3 . and the dimension of a column space actually has a specific term for it , and that 's called the rank . | hi sal , whats the difference between the dimension of the column space , which c ( a ) =dim ( a ) =3 where 3 is the dimension of basis , and the dimension of the rank ( a ) plus the null ( a ) , which is rank ( a ) + null ( a ) = 5 where 5 is the dimension of the matrix ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | well , the truth is , that the glucose ends up oftentimes in fruits and vegetables that we eat . but as far as the oxygen goes , it makes an excess of oxygen . so there is actually enough oxygen to go both to us , or to jack and to be used by itself . | what percentage goes to jack and what percentage goes to the plant ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | i 'm going to put a big plus sign , and i might even circle it because it 's so important . i do n't want you to lose track of it . and on the other side , of course , jack is getting something as well . | atp stands for adenine triphosphate , does n't it ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | and i 'm actually going to take just a moment to show you that this is n't the full story . there 's actually something else going on as well . and that is that there 's actually some cellular respiration happening on the plant 's side . | just to clear something , what is celluar respiration ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | and on the other side , of course , jack is getting something as well . he 's getting chemical energy . in fact , he 's using the chemical energy to help him climb the beanstalk . | how is jack getting chemical energy if chemical enery if for objects ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | this is our air breakdown . and carbon dioxide makes up about less than 1 % . so , that leaves you wondering , what the heck is making up all that other parts of air ? | why do we worry about global warming if there is less than 1 % carbon dioxide in the atmosphere ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | so on the one hand , you 've got what ? you 've got water because , of course , the beanstalk needs water , and you 've got carbon dioxide . and i 'm going to do carbon oxide in orange . | he seems to indicate that humans give of water ; how is this water given off ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | and in fact , if it takes in light energy right here , it needs to find a way to actually , eventually get some chemical energy itself , so that it can do all the things it needs to do . it does n't need to run because plants do n't move in that sense , but it might need to make new roots , and may need to make a flowe... | why we need plants to breath in ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | we have a planet full of humans , and full of other animals , and full of plants . what would the atmosphere look like ? this is the atmosphere . | how would it affect our lungs ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | that if you actually look at air , if you actually break down the atmosphere or air -- i 'm just going to write `` air '' here -- if you actually break it down , turns out that the ratios are actually a little different . so for example , oxygen makes up about 21 % of our air . this is our air breakdown . | can humans survive when oxygen content in the atmosphere is more than 21 % ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | it looks more filled with nitrogen than anything else . and in terms of carbon dioxide , it 's just got a little smidge of carbon dioxide . maybe right there . | why is there much more oxygen than carbon dioxide in the atmosphere ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | because everything is nice and balanced . and you can see how it makes perfect sense that , not only did jack need the beanstalk , but actually it sounds like the beanstalk needed jack , based on how i 've drawn it . now remember , none of this would even happen if there was no sunlight . so we actually need light ener... | would there be a reciprocal exchange of atoms between jack and the beanstalk ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | now remember , none of this would even happen if there was no sunlight . so we actually need light energy . in fact , that 's the whole purpose of this , right ? | i get that the moon has light but its important to delight the septerms that yes the moon has light but may i be remind-able that the moon has low light and it gets light from the sun so tell me ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | so nitrogen is part of us and is part of many , many living things . but nitrogen gas , specifically , is actually n2 . and n2 , this nitrogen gas , really is not too reactive . it kind of just hangs out by itself , does not like to react with things . so , looking at our little atmosphere graph , if you want to now th... | why does nitrogen not 'like to react ' with other chemicals ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | in fact , he 's using the chemical energy to help him climb the beanstalk . and so the chemical energy comes in the form of what we call atp , which is just a molecule of high energy . and so jack and the beanstalk are basically going from light energy to chemical energy using these two equations . | what is atp ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | it 's going to put out what ? oxygen and glucose . so i 'll put glucose up top and oxygen down below . | what happens with the glucose in plants that we 'do n't eat ' ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | it 's going to put out what ? oxygen and glucose . so i 'll put glucose up top and oxygen down below . | do plants use oxygen during photosynthesis ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | and in fact , if it takes in light energy right here , it needs to find a way to actually , eventually get some chemical energy itself , so that it can do all the things it needs to do . it does n't need to run because plants do n't move in that sense , but it might need to make new roots , and may need to make a flowe... | why do plants need photosynthesis to grow ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | there 's actually something else going on as well . and that is that there 's actually some cellular respiration happening on the plant 's side . so remember , not only does the human , or the jack , need energy , but so does the plant . | what is the scientific name for cellular respiration ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | in fact , that 's the whole purpose of this , right ? getting energy . so you have to have some light energy . | can plants use fats for their energy purposes ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | it 's going to put out what ? oxygen and glucose . so i 'll put glucose up top and oxygen down below . | how does our lung separate all those nitrogen from oxygen ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | it 's going to put out what ? oxygen and glucose . so i 'll put glucose up top and oxygen down below . | does n't the plant need glucose , and oxygen is just waste that is expelled in the process ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | it 's going to put out what ? oxygen and glucose . so i 'll put glucose up top and oxygen down below . | when you said jack was getting glucose and oxygen he was n't actually getting glucose at the time right ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | and there 's this really kind of interesting symbiosis . and by that i just mean that the two are kind of relying on each other to really work . so you kind of need both of them to work well . | what does atp mean ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | it does n't need to run because plants do n't move in that sense , but it might need to make new roots , and may need to make a flower , and all these things take energy . so actually , photosynthesis is happening during the day , but at all times plants are also capable of doing cellular respiration , just like humans... | when plants die , where do they go ? |
there 's a classic story out there and it has to do with a character named jack . and you may have heard this story , but i 'm sure that there is parts of that story that you have not heard . and so i 'm actually going to just try to fill in those parts that you get a complete idea of what happened . now jack came acro... | now remember , none of this would even happen if there was no sunlight . so we actually need light energy . in fact , that 's the whole purpose of this , right ? | do plants actually need co2 to live ? |
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