Let be a linear transformation in by .

1)Give the matrix of with respect to the canonical basis .

2)Tell whether is invertible or no and if it is calculate .

3)Give the matrix of with respect to the basis .

4)Give the matrix of T with respect to the basis , .

5)Give the matrix of with respect to the basis , .

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My attempt :

1) .

2)As is invertible, so is . I calculated which gave me .

3) I don't know how to do it! I guess I have to express the vectors of the basis of B as linear combination of the canonical vectors... but I'll have 3 scalars and I don't know what to do with them.

4)I think they ask for .

I wrote and I reduced the left matrix. At last the right matrix is the one they ask for and I got it to be .

5) If my attempt for 4) is good, I know how to do this one.

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I really need help for part 3), and I'd like to know if I did well what I did. Thanks in advance.