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Let $V,W$ finite dimensional vector spaces over the field $F$, and let $T:V\rightarrow W$ a linear transformation.

Define $T^* : W^* \rightarrow V^*$ the dual linear transformation. i.e $T^*(\varphi)=\varphi \circ T$.

We need to show that:

1) $([T]^{B}_{C})^t=[T^*]^{C^*}_{B^*}$ where $B=\{v_1,...,v_n\}$ is a basis of $V$, $C=\{w_1,...,w_n\}$ is a basis of $W$ and $B^*=\{\varphi _1,...,\varphi _n\}$ dual basis of $B$ and $B^*=\{\psi _1,...,\psi _n\}$ dual basis of $C$

2) Show that the correspondence $^ * :L(V,W) \to L({W^ * },{V^ * })$ defined by $T \mapsto {T^ * }$ is a vector space isomorphism.

I'm trying to show those properties without any success. 1) seems very technical, and in 2) it's suffice to show that the correspondence is injective (since the dimension are equal).

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    $\begingroup$ If $\;T:V\to W\;$ and $\;\phi\in V^*\;$ , as you defined, then $\;T^*:=\phi\circ T\;$ makes no sense. I think you meant $\; T:W^*\to V^*\;$ ? $\endgroup$ – Timbuc Apr 10 '15 at 13:11
  • $\begingroup$ 1) is in fact technical, though not too bad. You basically have to unfold everything into summation notation, then use commutativity, associativity, etc. 2) will actually be immediate from 1), provided you've already proven an isomorphism between matrices and linear maps. $\endgroup$ – GPerez Apr 10 '15 at 13:11
  • $\begingroup$ @Timbuc yes, I corrected it, thanks $\endgroup$ – Astro Nauft Apr 10 '15 at 13:13
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Let $A = \left(a_{ij}\right)$ be the matrix of $T$ with respect to the bases $\{v_1, \dots v_n\}$ of $V$ and $\{w_1, \dots w_n\}$ of $W$. By definition, this means that $\phi(v_j) = \sum_{i=1}^n a_{ij}w_i$ for $j=1, \dots , n$. To show that the matrix of $T^*$ is $A^T$, we need to show that $T^*(\psi_j) = \sum_{i=1}^n a_{ji} \phi_i$. For each $k = 1, \dots , n$, we have \begin{align*} T^*(\psi_j)(v_k) &= \psi_j(T(v_k))\\ &= \psi_j \left( \sum_{i=1}^n a_{ik} w_i \right)\\ &= \sum_{i=1}^n a_{ik} \psi_j(w_i)\\ &= a_{jk}\\ &= \sum_{i=1}^n a_{ji}\phi_i(v_k)\\ &= \left(\sum_{i=1}^n a_{ji}\phi_i\right)(v_k), \end{align*} so $T^*(\psi_j)$ and $\sum_{i=1}^n a_{ji} \phi_i$ agree on a basis of $V$ and are therefore equal.

The second part then follows as GPerez remarked.

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