Invertible matrix($T:M_{2*2}(R) \rightarrow M_{2*2}(R)$ defined by T( $\bigl( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \bigr)$

As @user1551 mentioned in the comments, you may apply the stated theorem. Alternatively, one may observe that $$ T\begin{pmatrix} b & a-b\\ c & d-c \end{pmatrix} = \begin{pmatrix} a & b\\ c & d \end{pmatrix}, $$

to see that $T$ is onto.

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Linear Algebra

Linear Transformations

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