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Solution to Linear Algebra Done Wrong Exercise 1.2.1


Find a basis in the space of $3\times 2$ matrices $M_{3\times 2}$.


Solution: Any $3\times 2$ matrix $A$ has the form
$$ A=\begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \\ a_3 & b_3 \end{pmatrix} $$ Define six $3\times 2$ matrices as follows, $$ {\bf e}_1=\begin{pmatrix} 1 & 0 \\ 0 & 0 \\ 0 & 0 \end{pmatrix}\quad {\bf e}_2=\begin{pmatrix} 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{pmatrix} $$ $$ {\bf e}_3=\begin{pmatrix} 0 & 0 \\ 1 & 0 \\ 0 & 0 \end{pmatrix} \quad {\bf e}_4=\begin{pmatrix} 0 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix} $$ $$ {\bf e}_5=\begin{pmatrix} 0 & 0 \\ 0 & 0 \\ 1 & 0 \end{pmatrix} \quad {\bf e}_6=\begin{pmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 1 \end{pmatrix} $$ Then the matrix $A$ can be uniquely written as $$
A= a_1 {\bf e}_1+ b_1 {\bf e}_2+ a_2 {\bf e}_3+ b_2 {\bf e}_4+ a_3 {\bf e}_5+ b_3 {\bf e}_6.
$$ Hence ${\bf e}_1, {\bf e}_2, {\bf e}_3, {\bf e}_4, {\bf e}_5, {\bf e}_6$ is a basis in the space of $3\times 2$ matrices $M_{3\times 2}$.


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