Split (1)

train · 3.69k rows

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ZK3O402wf1c	For me, this matrix multiplication
ZK3O402wf1c	says I take one of that column and two of that column and add.
ZK3O402wf1c	So this is the way I would think of it
ZK3O402wf1c	is one of the first column and two of the second column
ZK3O402wf1c	and let's just see what we get.
ZK3O402wf1c	So in the first component I'm getting a two and a ten.
ZK3O402wf1c	I'm getting a twelve there.
ZK3O402wf1c	In the second component I'm getting a one and a six,
ZK3O402wf1c	I'm getting a seven.
ZK3O402wf1c	So that matrix times that vector is twelve seven.
ZK3O402wf1c	Now, you could do that another way.
ZK3O402wf1c	You could do it a row at a time.
ZK3O402wf1c	And you would get this twelve -- and actually I pretty much did
ZK3O402wf1c	it here --
ZK3O402wf1c	this way.
ZK3O402wf1c	Two -- I could take that row times my vector.
ZK3O402wf1c	This is the idea of a dot product.
ZK3O402wf1c	This vector times this vector, two times one plus five times
ZK3O402wf1c	two is the twelve.
ZK3O402wf1c	This vector times this vector -- one times one plus three times
ZK3O402wf1c	two is the seven.
ZK3O402wf1c	So I can do it by rows, and in each row times
ZK3O402wf1c	my x is what I'll later call a dot product.
ZK3O402wf1c	But I also like to see it by columns.
ZK3O402wf1c	I see this as a linear combination of a column.
ZK3O402wf1c	So here's my point.
ZK3O402wf1c	A times x is a combination of the columns of A.
ZK3O402wf1c	That's how I hope you will think of A times x when we need
ZK3O402wf1c	it.
ZK3O402wf1c	Right now we've got -- with small ones,
ZK3O402wf1c	we can always do it in different ways, but later,
ZK3O402wf1c	think of it that way.
ZK3O402wf1c	Okay.
ZK3O402wf1c	So that's the picture for a two by two system.
ZK3O402wf1c	And if the right-hand side B happened to be twelve seven,
ZK3O402wf1c	then of course the correct solution would be one two.
ZK3O402wf1c	Okay.
ZK3O402wf1c	So let me come back next time to a systematic way,
ZK3O402wf1c	using elimination, to find the solution,
ZK3O402wf1c	if there is one, to a system of any size and
PxCxlsl_YwY	So let's start right away with stuff that we will need to see
PxCxlsl_YwY	before we can go on to more advanced things.
PxCxlsl_YwY	So, hopefully yesterday in recitation, you heard a bit
PxCxlsl_YwY	about vectors. How many of you actually knew
PxCxlsl_YwY	about vectors before that? OK, that's the vast majority.
PxCxlsl_YwY	If you are not one of those people, well,
PxCxlsl_YwY	hopefully you'll learn about vectors right now.
PxCxlsl_YwY	I'm sorry that the learning curve will be a bit steeper for
PxCxlsl_YwY	the first week. But hopefully,
PxCxlsl_YwY	you'll adjust fine. If you have trouble with
PxCxlsl_YwY	vectors, do go to your recitation instructor's office
PxCxlsl_YwY	hours for extra practice if you feel the need to.
PxCxlsl_YwY	You will see it's pretty easy. So, just to remind you,
PxCxlsl_YwY	a vector is a quantity that has both a direction and a magnitude
PxCxlsl_YwY	of length.
PxCxlsl_YwY	So -- So, concretely the way you draw a vector is by some
PxCxlsl_YwY	arrow, like that, OK?
PxCxlsl_YwY	And so, it has a length, and it's pointing in some
PxCxlsl_YwY	direction. And, so, now,
PxCxlsl_YwY	the way that we compute things with vectors,
PxCxlsl_YwY	typically, as we introduce a coordinate system.
PxCxlsl_YwY	So, if we are in the plane, x-y-axis, if we are in space,
PxCxlsl_YwY	x-y-z axis. So, usually I will try to draw
PxCxlsl_YwY	my x-y-z axis consistently to look like this.
PxCxlsl_YwY	And then, I can represent my vector in terms of its
PxCxlsl_YwY	components along the coordinate axis.
PxCxlsl_YwY	So, that means when I have this row, I can ask,
PxCxlsl_YwY	how much does it go in the x direction?
PxCxlsl_YwY	How much does it go in the y direction?
PxCxlsl_YwY	How much does it go in the z direction?
PxCxlsl_YwY	And, so, let's call this a vector A.
PxCxlsl_YwY	So, it's more convention. When we have a vector quantity,
PxCxlsl_YwY	we put an arrow on top to remind us that it's a vector.
PxCxlsl_YwY	If it's in the textbook, then sometimes it's in bold
PxCxlsl_YwY	because it's easier to typeset. If you've tried in your
PxCxlsl_YwY	favorite word processor, bold is easy and vectors are
PxCxlsl_YwY	not easy. So, the vector you can try to
PxCxlsl_YwY	decompose terms of unit vectors directed along the coordinate
PxCxlsl_YwY	axis. So, the convention is there is
PxCxlsl_YwY	a vector that we call *amplt;iamp*gt;
PxCxlsl_YwY	hat that points along the x axis and has length one.
PxCxlsl_YwY	There's a vector called *amplt;jamp*gt;
PxCxlsl_YwY	hat that does the same along the y axis,
PxCxlsl_YwY	and the *amplt;kamp*gt;
PxCxlsl_YwY	hat that does the same along the z axis.
PxCxlsl_YwY	And, so, we can express any vector in terms of its
PxCxlsl_YwY	components. So, the other notation is
PxCxlsl_YwY	*amplt;a1, a2, a3 amp*gt;
PxCxlsl_YwY	between these square brackets. Well, in angular brackets.
PxCxlsl_YwY	So, the length of a vector we denote by, if you want,
PxCxlsl_YwY	it's the same notation as the absolute value.
PxCxlsl_YwY	So, that's going to be a number, as we say,
PxCxlsl_YwY	now, a scalar quantity. OK, so, a scalar quantity is a
PxCxlsl_YwY	usual numerical quantity as opposed to a vector quantity.
PxCxlsl_YwY	And, its direction is sometimes called dir A,
PxCxlsl_YwY	and that can be obtained just by scaling the vector down to
PxCxlsl_YwY	unit length, for example,
PxCxlsl_YwY	by dividing it by its length. So -- Well, there's a lot of
PxCxlsl_YwY	notation to be learned. So, for example,
PxCxlsl_YwY	if I have two points, P and Q, then I can draw a

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