Question

Consider an n$\mathbf{\times }$m matrix A, a vector in ${\mathbf{R}}^{\mathbf{m}}$, and a vector $\overrightarrow{\mathbf{W}}$in ${\mathbf{R}}^{\mathbf{n}}$. Show that $\mathbf{\left(}\mathbf{A}\overrightarrow{\mathbf{v}}\mathbf{\right)}\mathbf{\cdot }\overrightarrow{\mathbf{W}}\mathbf{=}\overrightarrow{\mathbf{V}}\mathbf{\cdot }\left(A^T\overrightarrow{w}\right)$.

Short answer

It is proved that $\left(\mathrm{A}\overrightarrow{\mathrm{v}}\right)\cdot \overrightarrow{\mathrm{W}}=\overrightarrow{\mathrm{V}}\cdot \left({\mathrm{A}}^{\mathrm{T}}\overrightarrow{\mathrm{w}}\right)$.

Step-by-step solution

Consider the theorem

If $\overrightarrow{\mathbf{v}}$and $\overrightarrow{\mathbf{w}}$are two (column) vectors in${\mathbf{R}}^{\mathbf{n}}$, then$\begin{array}{r}\overrightarrow{\mathbf{V}}\mathbf{\cdot }\overrightarrow{\mathbf{w}}\mathbf{=}{\overrightarrow{\mathbf{V}}}^{\mathbf{\top }}\overrightarrow{\mathbf{w}}\\ \mathbf{\uparrow }\mathbf{\uparrow }\\ \underset{\mathbf{\text{product}}}{\underbrace{\mathbf{\left(}\mathbf{\text{dot}}\mathbf{\right)}\mathbf{(}\mathbf{\text{matrix)}}}}\end{array}$

Application of the theorem

Observe that,

According to the theorem.

$\begin{array}{r}\left(\mathrm{A}\overrightarrow{\mathrm{v}}\right)\cdot \overrightarrow{\mathrm{w}}=(\mathrm{A}\overrightarrow{\mathrm{V}}{)}^{\mathrm{T}}\overrightarrow{\mathrm{w}}(\text{Theorem 5.3.6)}\\ =\left({\overrightarrow{\mathrm{v}}}^{\mathrm{T}}{\mathrm{A}}^{\mathrm{T}}\right)\overrightarrow{\mathrm{w}}\left(\text{Theorem5.3.9}\right(\mathrm{c}\left)\right)\\ ={\overrightarrow{\mathrm{v}}}^{\mathrm{T}}\left({\mathrm{A}}^{\mathrm{T}}\overrightarrow{\mathrm{w}}\right)\left(\text{Associativity}\right)\\ =\overrightarrow{\mathrm{v}}\cdot \left({\mathrm{A}}^{\mathrm{T}}\overrightarrow{\mathrm{w}}\right)\end{array}$

Hence, $\left(\mathrm{A}\overrightarrow{\mathrm{v}}\right)\cdot \overrightarrow{\mathrm{W}}=\overrightarrow{\mathrm{V}}\cdot \left({\mathrm{A}}^{\mathrm{T}}\overrightarrow{\mathrm{w}}\right)$is proved.