Matrices and Rotation. Matrix Multiplication

Matrices and Rotation. Matrix Multiplication

Let's begin by discussing matrices. You should remember how to multiply a matrix times a
vector.
If we have a matrix, say a, b, c, d (or these could be numbers) and multiply it by a vector v,
v1, v2 (or these could be numbers), what do we get then? A new vector.
For the top component, we multiply times v1 by a plus b times v2. Somehow, if you put
my fingers like this, it helps me remember what to do.

For the bottom component, you
multiply across the bottom row: a times v1 by a plus b times v2. And you do c times v1
plus d times v2.
When you first learn this formula, you may have a thought like, "What is so important about
this formula in particular?" The reason is that you will see formulas that look like this all
over the place in science and engineering.
We're looking for a number of times v1 plus a number of times v2 for the first component,
and a number of times v1 plus a number of times v2 for the second component.

If you look for it, you'll see it often. There is an example on the board. Where do we see
over here?
So choosing theta pi over 6, this is a number of times v1 plus a number of times v2. So
that's the first component, and in this case, we have a matrix
Whenever we see that, we know that a matrix could be used to represent it.

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The matrix is implicit when we write this. Let's write it down. Where is the best place to put
it? Probably we can fit it here. This was not planned out, but we should have written at the
very top.
Let's write this in matrix form. w1, w2 is our matrix, times v1, v2. So w1 is cosine of theta
v1– so we have cosine of theta here– minus sine of theta v2. Remember to check.
We want w1 to be the cosine of theta v1 minus sine of theta v2 and we want w2 to be this
one. So it's a sine of theta v1 and cosine of theta v2. That is the matrix.
As we keep our eyes open, we will notice many problems in various parts of science and
engineering, other areas of math that can be written this way. And then they can all be
expressed as matrices.
Thus, it is worth noting that all these problems are similar to each other, because whatever
we figured out about the first problem can be applied to another problem in another area.
There are many tools for dealing with matrices, which either goes by the name linear
algebra or is called a matrix.
And then, when we recognize that a problem is coming from a matrix, we can use all of the
tools in the matrix toolbox to address it. Maybe we won't cover all of those tools, but you
will study one or two or three of them.
We will see a few examples of problems that can be represented by matrices, and we will
try to use the tool on the problem to see what we can figure out. We'll explore some
examples of problems that can be written using matrices, and then use the methods we've
learned to solve them. That is our plan for studying matrices.
Matries
(((a,b),(c,d))((v_1)(v_2))=((av_1+bv_2),(cv_1+dv_2))
((w_1),(w_2))=((cosθ+sinθ),(sinθ+cosθ))((v_1),(v_2))