Matrix question

Sorry about the notation.

Find a,b,c,d such that


| a b |
| c d | < x, y> = <y, x>

 

Let us think about what matrix multiplication encodes. It is linear, so to understand what it does you just need to understand what it does to a basis (i.e. A(x+y)=Ax + Ay, so knowing what happens to x and y tells you what happens to x+y).

What happens if I multiply a matrix A into the column vector (1,0)^t? I get out the first column of A (check this!). Now find out what you get from multiplying into (0,1)^t

So if I know that A sends (1,0)^t to c and (0,1)^t to d, then I can write down A's matrix immediately.

 

Here is the simplest way to find the matrix of a linear tranformation, in a given basis.

Apply the transformation to the first basis vector: write the result in terms of the basis. The coefficients are the first column in the matrix.

Now do the same to the second basis vector. That gives the second column.

I notice, now, that this is exactly what matt grime said!

That is, write the matrix, as you do:
[tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}[/tex]
and apply it to [1 0].
[tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}1 \\ 0\end{bmatrix}= \begin{bmatrix}a \\ c\end{bmatrix}[/tex]
But since this transformation "swaps the two coordinates", [1 0] is taken to [0 1] so you must have [a c]= [0 1].