Similarity among nonsquare matrices

Let F be a field, let V be an n-dimensional vector space over F, and let W be an m-dimensional vector space over F. Let B_1, B_2 \subseteq V and E_1, E_2 \subseteq W be bases, let \varphi : V \rightarrow W be a linear transformation, and let A = M^{E_1}_{B_1}(\varphi) and B = M^{E_2}_{B_2}(\varphi). Let P = M^{B_1}_{B_2}(1) be the matrix of the identity map on V with respect to the source basis B_2 and the target basis B_1, and likewise let Q = M^{E_1}_{E_2}(1) be the matrix of the identity map on W with respect to the source basis E_2 and the target basis E_1.

Prove that Q^{-1} = M^{E_2}_{E_1}(1) and that Q^{-1}AP = B.


Using Theorem 12 in D&F, we have Q \times M^{E_2}_{E_1}(1) = M^{E_1}_{E_2}(1) \times M^{E_2}_{E_1}(1) = M^{E_1}_{E_1}(1 \circ 1) = M^{E_1}_{E_2}(1) = I. Similarly, M^{E_2}_{E_1}(1) \circ Q = M^{E_2}_{E_2}(1) = I. By the uniqueness of inverses, M^{E_2}_{E_1}(1) = Q^{-1}.

Now Q^{-1}AP = M^{E_2}_{E_1}(1) \times M^{E_1}_{B_1}(\varphi) \times M^{B_1}_{B_2}(1) = M^{E_2}_{B_2}(1 \circ \varphi \circ 1) = M^{E_2}_{B_2}(\varphi) = B.

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