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I am having difficulty with this question and would appreciate any help.

Let $T: V \rightarrow W$ be a linear transformation from the vector space $V$ to the vector space $W$. Let $B$ be a basis for $V$ and let $C$ be a basis for $W$. Let $[T]_{C\leftarrow B}$ be the matrix for $T$ relative to the bases $B$ and $C$. Let $r$ be the rank of $[T]_{C\leftarrow B}$.

A) Prove that there exists a basis $D$ of $W$ such that $[T]_{D\leftarrow B}$ has exactly $r$ nonzero rows.

B) Prove that $r=\dim(\operatorname{range}(T))$.

C). Let $V=\mathbb{R}^2$ and $W=\mathbb{R}^3$ and suppose $[T]_{C\leftarrow B}= \left[\begin{array}{cc} 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ \end{array} \right]$. Prove that there does not exist a basis $A$ of $V$ such that $[T]_{C\leftarrow A}$ has exactly two nonzero rows.

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Can you give your question a more descriptive title? –  Kazark Mar 19 '13 at 18:44
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1 Answer

Hints:

A) Consider a basis $D$ such that $[T]_{D\leftarrow B}$ is a row reduced echelon form of $[T]_{C\leftarrow B}$.

B) With respect to the bases of $B$ and $D$, $\operatorname{range}(T)$ is represented by the column space of $[T]_{D\leftarrow B}$. Now consider the columns containing the pivots in the row reduced echelon form $[T]_{D\leftarrow B}$.

C) Let $C=\{w_1,w_2,w_3\}$. If such a basis $A$ exists, show that for some $w_i\in C$, $Tv$ is always a scalar multiple of $w_i$. However, by considering $[T]_{C\leftarrow B}$, argue that this is impossible. Alternatively, show that by changing basis in $V$, the column space of the matrix of $T$ remains unchanged.

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