Is the Inner Product of Matrices Preserved under Vector Multiplication?

[Chris L T521]

Thanks to those who participated in last week's POTW! Here's this week's problem!

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Problem: Let $A$ and $B$ be $n\times n$ matrices with real entries. Show that $\langle A\mathbf{u},B\mathbf{v}\rangle = \langle \mathbf{u}, A^TB\mathbf{v}\rangle$ for any vectors $\mathbf{v},\mathbf{w}\in\mathbb{R}^n$, where $\langle\cdot,\cdot\rangle$ denotes the standard inner product on $\mathbb{R}^n$.

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[Chris L T521]

This week's question was correctly answered by dwsmith, girdav, and Sudharaka. You can find Sudharaka's solution below.

Spoiler

Take any two vectors \(\mathbf{x},\mathbf{y}\in\mathbb{R}^n\) where \(\mathbf{x}=(x_1,x_2,\cdots,x_n)^{T}\) and \(\mathbf{y}=(y_1,y_2,\cdots,y_n)^{T}\). Then the standard inner product of \(\mathbb{R}^n\) is defined by,

\[\langle\mathbf{x},\mathbf{y}\rangle:=\sum_{i=1}^{n}x_{i}y_{i}=x^{T}y\]Hence we have,\[\langle A\mathbf{u},B\mathbf{v}\rangle =(A\mathbf{u})^{T}(B\mathbf{v})=(\mathbf{u}^{T}A^{T})(B\mathbf{v})~~~~~~~~(1)\]Also,\[\langle \mathbf{u}, A^TB\mathbf{v}\rangle=\mathbf{u}^{T}(A^{T}B\mathbf{v})=(\mathbf{u}^{T}A^{T})(B\mathbf{v})~~~~~~~(2)\]By (1) and (2) we have,\[\langle A\mathbf{u},B\mathbf{v}\rangle = \langle \mathbf{u}, A^TB\mathbf{v}\rangle\]