matrices and mappings?

Question

tell me everything you know about how to map points on a plane using matrices. NOW!!! lol

Answer 1

Everything I know about this is probably too much, but I'll tell you some. My answer to your previous question (thankfully, more specific than this one!!!) tells you how to transform a point using a matrix, so I won't go over that again.

Using the matrix
[a .. b]
[c .. d]
to transform the point (1, 0) gives
[a .. b] * [1]
[c .. d] .. [0]

= [a]
.. [c]
i.e. the first column of the matrix is the point that (1, 0) is transformed to.
Transforming (0, 1) gives
[b]
[d]
i.e. the second column is the transformation of (0,1)
so you can find the matrix for any transformation by finding the images of (1,0) -- that gives first column -- and (0, 1) for the second column.

For example, if we rotate through 90 degrees, (1,0) on the x axis goes to (0,1) on the y axis, so first column is (0,1). Also (0,1) on the y axis turns around to (-1,0) on the negative part of the x axis, so that gives the second column. We can use this matrix to rotate any figure through 90 degrees.

e.g. Take the triangle (3,1) (-1,2) (1, -4). Write these coordinates as column vectors, and multiply them by the matrix operator found in the previous paragraph:
[0 ..-1] * [3 .. -1 .. 1]
[1 .. 0] .. [1 ... 2 ..-4]

= [-1 ..-2 .. 4]
...[3 .. -1 .. 1]
and so the vertices of the rotated triangle are
(-1,3) (-2, -1) (4, 1).

Let me show the first point in detail to make sure you know what I'm doing:
row1 * column1 = 0*3 + -1*1 = -1
row2 * column 1 = 1*3 + 0*1 = 3
so that's how I got (-1, 3) in the first column. You work out the other two.

Try putting all this on a sketch, and you'll see it beautifully. It's fun!

Answer 2

I know of using vectors in R2 to do that: