How to find a matrix, with respect to a basis

[bonappler]

I hope I don't seem greedy but I need help with this other question. I'm not exactly sure how to find a matrix with respect to a basis. In my mind I think I can do it using simultaneous equations but I am not sure if this is correct. I'll be happy to take a picture of my working out and post it.

 

[stapel]

The image is unreadable, at least for me. Please reply with either a better graphic or else a typed-out version of the exercise. Note: You can use "code" tags to keep things neat, if you like:

A = [ 1 2 3 ]
    [ 4 5 6 ]
    [ 7 8 9 ]

 

[HallsofIvy]

I also cannot read your image but in general:

To find the matrix representation of a linear transformation in a given ordered basis,
1) Apply the linear transformation to each basis vector in turn.
2) Write the result as a linear combination of the basis vectors.

The coefficients in the linear combination give the columns of the matrix representation.

For example, if L<a, b>= <a- b, 2b+ 3a> then to find its matrix representation in the basis u= <1, 1> , v=<-1, 1>
1) Lu= <1- 1, 2+ 3>= <0, 5>
2) <0, 5>= (5/2)<1, 1>+ (5/2)< -1, 1>
so the first column is \(\displaystyle \begin{bmatrix} \frac{5}{2} \\ \frac{5}{2}\end{bmatrix}\)

1) Lv= <-1- 1, 2- 3>= <-2, -1>
2) <-2, -1>= (-3/2)<1, 1>+ (1/2)< -1, 1>
so the second column is \(\displaystyle \begin{bmatrix}\frac{-3}{2} \\ \frac{1}{2}\end{bmatrix}\)

The matrix representing L in the basis {<1, 1>, <-1, 1>} is
\(\displaystyle \begin{bmatrix}\frac{5}{2} & -\frac{3}{2} \\ \frac{5}{2} & \frac{1}{2} \end{bmatrix}\)