Finding the matrix with respect to two bases

[Niles]

Homework Statement

If I am giving a matrix A representing the linear transformation L(x) = Ax, then A is the matrix-representation of L with respect to the basis elements in the vector-space S (standard).

If I am now given another vectorspace V by the matrix V, and I want to find the matrix B representing L with respect to the bases S and V, I use:

B = V^(-1) * S.

The Attempt at a Solution

Is what I wrote correct?

Thanks in advance.

Sincerly, Niles.

 

[HallsofIvy]

Homework Statement

If I am giving a matrix A representing the linear transformation L(x) = Ax, then A is the matrix-representation of L with respect to the basis elements in the vector-space S (standard).

If I am now given another vectorspace V by the matrix V, and I want to find the matrix B representing L with respect to the bases S and V, I use:

B = V^(-1) * S.

The Attempt at a Solution

Is what I wrote correct?

Thanks in advance.

Sincerly, Niles.

I don't know what you mean by "given another vectorspace V by the matrix V". First, since you were able to write L as a matrix using one basis for S, the L must be a transformation from S to itself- you can't be "given another vector space V". Second, since a vectorspace and a matrix are different objects, "vectorspace V" and "matrix V" makes no sense. I suspect you mean that you are given a new basis (as columns of a matrix?) for the same vectorspace S. The standard way of representing a linear transformation from U to V, as a matrix, for given bases of U and V is: Apply the transformation to each basis vector of U in turn. Right the result in terms of basis V. The coefficients give a column of the matrix.

For example, suppose L:R²-> R² takes basis vector <1, 0> to <4, 2>. If {<1, 1>, <1, -1>} is another basis, then <4, 2>= 3<1,1>+ 1<1,-1>. The first column is <3, 1>.