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Let $T: V \to W$ be a linear transformation of two finite dimensional vector spaces $V,W$ (both over a field $F$).

I'm also given a transformation matrix $A$.

My assignment is to show that there exists a basis $B$ in $V$ and a basis $C$ in $W$ such that the transformation $F$ with respect to the bases $B$ and $C$ actually does have $A$ as a transformation matrix.

I'm asking this question in a kind of general sense (i.e not giving you $A$) so that I can solve it myself and learn as much as possible. But need some help kind of parsing the question and finding a strategy for finding a solution.

Since $V,W$ are vector spaces of course they have bases and I can do a transformation with respect to the bases from $V$ to $W$. But how do I show that (some specific?) bases exist such that the transformation with respect to these bases have a given a $A$ as a transformation matrix?

One naive idea:

Assume $A$ is transformation matrix of $T$. Then we take one vector in $V$ expressed in (some basis) $B$ and when we transform it to some vector in $W$, expressed in (some basis) $C$. So $A$ does at least change the basis of a vector, but I since $V$ and $W$ might be of different dimension I don't really know where to go from here. Am I one the wrong track?

What can I say about the existence of bases if I assume that A is in fact the transformation matrix?

Or should I rather "bruteforce" it by trying out pairs general bases and hope that I stumble upon $A$?

Edit:

I have to admit I'm stuck. I can't solve it. My given transformation matrix is: $\begin{bmatrix} I &0 \\ 0& 0 \end{bmatrix}$, where I is the identity matrix "of a certain size". Can anyone perhaps help me out?

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Hint: assume $\dim V = n$ and $\dim W = n$, then, by Steinitz's Theorem, $V$ and $W$ have bases with $n$ and $m$ elements respectively. Elements which you can write as the columns of two matrices, say $B$ and $C$. Now, think about how $B$ and $C$ are and how do they interact with $A$.

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  • $\begingroup$ I have to admit I'm stuck. I can't solve it. My given transformation matrix is: $\begin{bmatrix} I &0 \\ 0& 0 \end{bmatrix}$, where I is the identity matrix "of a certain size". Can anyone perhaps help me out? $\endgroup$ – John Smith Jan 26 '14 at 12:50

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