Matrix m(T)^F_E Explained: Linear Maps U & V

[sunnyday11]

Homework Statement

Let T: U-->V be a linear map between vector spaces U and V and let E be basis for U and F be a basis for V. Explain what is meant by the matrix m(T)^{F}_{E} of T taken with respect to E on the left and F on the right.

Homework Equations

The Attempt at a Solution

I said it means T(E) = \sum^{n}_{j=1} a_jif_j

where M(T)^{F}_{E} = a_ji

But the marker said describe matrix and I'm not quite sure what to describe.

 

[HallsofIvy]

The columns of the matrix are the coefficients of the vectors T(u_i), where u_i are the vectors in basis E written as a linear as a linear combination of the vectors in basis F. that is, I think, essentially what you wrote except that "T(E)" makes no sense to me.