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Let $V$ and $W$ be finite-dimensional vector spaces with $\dim(V)=\dim(W)$ and let $T:V \rightarrow W$ be a linear map. How do I prove that there are bases $\beta$ of $V$ and $\gamma$ of $W$ such that the matrix $[T]^{\gamma}_{\beta}$ is diagonal?

\begin{eqnarray} v_1\mapsto a_{11}w_1\\ v_2\mapsto a_{22}w_2\\ v_3\mapsto a_{33}w_3\\ \vdots\hspace{1cm} \end{eqnarray}

Right?

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2 Answers 2

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Let $V'$ an additional subspace of $\ker T$ in V i.e. such that $$V=\ker V\oplus V'$$ then we know that $T_{|V'}:V'\to \mathrm{Im T}$ is an isomorphism and let $(e_1,\ldots,e_r)$ a basis for $V'$ and $(e_{r+1},\ldots,e_n)$ a basis for $\ker V$ then we have $\beta=(e_1,\ldots,e_n)$ is a basis for $V$ and we have $(T(e_1),\ldots,T(e_r))$ is a basis for $\mathrm{Im }T$ which we complete in a basis $\gamma=(T(e_1),\ldots,T(e_r),v_{r+1},\ldots,v_n)$ for $W$ hence \begin{eqnarray} [T]_{\beta}^{\gamma} = \begin{pmatrix} I_r & 0 \\ 0 & 0 \end{pmatrix} \end{eqnarray}

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  • $\begingroup$ @SamiBenRombhane We need to find bases which can interact in this way (see edit). $\endgroup$
    – Trancot
    Apr 25, 2013 at 20:59
  • $\begingroup$ @Trancot take $a_{11}=1,\ldots,a_{rr}=1$ and $a_{ii}=0$ for the other $\endgroup$
    – user63181
    Apr 25, 2013 at 21:13
  • $\begingroup$ Yes, but how is it shown that these bases exist? $\endgroup$
    – Trancot
    Apr 25, 2013 at 21:17
  • $\begingroup$ You must know that every vector space has a basis see en.wikipedia.org/wiki/Basis_(linear_algebra) $\endgroup$
    – user63181
    Apr 25, 2013 at 21:21
  • $\begingroup$ user63181, Why the restriction map $T|_{V^{\prime}}$ is isomorphism. I know $\text{dim}(V^{\prime})=\text{dim}(\text{Im} T)$, but why this is injective? $\endgroup$
    – MathBS
    Oct 9, 2018 at 19:13
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Choose any basis $\beta=\{b_1,\ldots,b_n\}$ of $V$. As I discussed in an answer to one of your earlier questions, $[T]_{\beta}^{\gamma}\in\mathrm{M}_{n\times n}(F)$ is the matrix $$\begin{bmatrix} a_{11} & \cdots & a_{1n}\\ \vdots & \ddots & \vdots\\ a_{n1} & \cdots & a_{nn} \end{bmatrix}$$ where the $a_{ij}$ are the elements of $F$ uniquely determined by $$T(b_i)=\sum_{j=1}^n a_{ij}\gamma_j.$$ What would the $\gamma_j$'s have to be to make the matrix diagonal? (Note: some of the $\gamma_j$'s might be up to you to choose, i.e. not constrained to be something particular, depending on what $T$ is.)

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  • $\begingroup$ Would they be $c_i\delta_{ij}$? $\endgroup$
    – Trancot
    Apr 25, 2013 at 17:53
  • $\begingroup$ @Trancot: Those are not elements of $W$. $\endgroup$ Apr 25, 2013 at 17:54
  • $\begingroup$ We need to find bases which can interact in this way (see edit). I don't know what would they be then? $\endgroup$
    – Trancot
    Apr 25, 2013 at 21:01

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