Answer
See solution
Work Step by Step
Put $T: \mathbb{R}^{n} \mapsto \mathbb{R}^{m}$ be a linear transformation such
that $T(\mathbf{x})=B \mathbf{x}$ for some $m \times n$ matrix.
Goal
Verify the uniqueness of $A$ in Theorem 10
Concepts
Theorem 10
Standard Matrix
Plan
Show that if $A$ is the standard matrix for $T,$ then
\[
A=B
\]
Show that $A$ and $B$ have the same columns.
Solve
Define $\mathrm{T}: \mathbb{R}^{n} \mapsto \mathbb{R}^{m}$ by $T(\mathrm{x})=B \mathrm{x}$ for some $m \times n$
matrix $\mathrm{B},$ and put $A$ be the standard matrix for $T$.
By definition, $A=\left[\begin{array}{lll}T\left(\mathbf{e}_{1}\right) & \ldots & T\left(\mathbf{e}_{j}\right)\end{array}\right]$ where $\mathbf{e}_{j}$
is the $j$ th column of $I_{n}$
\[
\text { By matrix-vector multiplication, } T\left(\mathbf{e}_{j}\right)=B \mathbf{e}_{j}=b_{j}
\]
the $j$ th column of $B .$ So $A=\left[\begin{array}{lll}\mathbf{b}_{1} & \ldots & \mathbf{b}_{2}\end{array}\right]=B$
Conclusion
Matrix $A$ is unique.
