The matrix of a nonsingular linear transformation is nonsingular, regardless of the bases chosen

Let F be a field, and let V and W be n– and m-dimensional vector spaces over F, respectively. Prove that, for any bases B \subseteq V and E \subseteq W, \varphi : V \rightarrow W is a nonsingular linear transformation if and only if M^E_B(\varphi) is nonsingular matrix.


Choose some basis B \subseteq V and E \subseteq W.

Suppose \varphi : V \rightarrow W is a nonsingular linear transformation. Then \mathsf{ker}\ \varphi = 0. Now suppose x \in V. If M^E_B(\varphi) \times x = 0, then \varphi(x) = 0, and so x = 0. Thus M^E_B(\varphi) is a nonsingular matrix.

Conversely, suppose M^E_B(\varphi) is a nonsingular matrix. Suppose x \in \mathsf{ker}\ \varphi. Then \varphi(x) = 0, and so we have M^E_B(\varphi) \times x = 0. Since M^E_B(\varphi) is nonsingular, x = 0. Thus \mathsf{ker}\ \varphi = 0, and so \varphi is nonsingular as a linear transformation.

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