?

Linear Algebra

Free Version

Upgrade subject to access all content

Difficult

Inner Product Space: Coming From Symmetric Matrices

LINALG-C9MW1A

Let $\langle\ ,\ \rangle : \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3$ be defined as:

$$\langle \boldsymbol{u}, \boldsymbol{v}\rangle = \boldsymbol{u}^TA\boldsymbol{v}$$

$\ldots$ for $\boldsymbol{u},\ \boldsymbol{v} \in \mathbb{R}^3$ and for some $3\times 3$ matrix $A$. Which of the following matrices make $\langle\ ,\ \rangle$ into an inner product.

Select ALL that apply.

A

$\begin{bmatrix} 1 & 2 & 3 \\\ 3 & 2 & 1 \\\ 2 & 1 & 3 \end{bmatrix}$

B

$\begin{bmatrix} 1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 1 \end{bmatrix}$

C

$\begin{bmatrix} 2 & 0 & 0 \\\ 0 & 2 & 0 \\\ 0 & 0 & 2 \end{bmatrix}$

D

$\begin{bmatrix} 1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 0 \end{bmatrix}$

E

$\begin{bmatrix} 1 & 2 & 3 \\\ 2 & 3 & 1 \\\ 3 & 1 & 2 \end{bmatrix}$

F

$\begin{bmatrix} 1 & -2 & 3 \\\ -2 & 3 & 1 \\\ 3 & 1 & -2 \end{bmatrix}$