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Linear Algebra

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Inner Product Space: Coming From Symmetric $2 \times 2$ Matrices

LINALG-Z8DW3C

Let $\langle\ ,\ \rangle : \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}^2$ be defined as:

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

$\ldots$ for $\boldsymbol{u},\ \boldsymbol{v} \in \mathbb{R}^2$ and for some $2\times 2$ matrix $A$. Which of the following matrices make $\langle\ ,\ \rangle$ into an inner product.

Select ALL that apply.

A

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

B

$\begin{bmatrix} 1 & 0 \\\ 0 & 2 \end{bmatrix}$

C

$\begin{bmatrix} 1 & 0 \\\ 0 & -2 \end{bmatrix}$

D

$\begin{bmatrix} 1 & 1 \\\ 0 & 2 \end{bmatrix}$

E

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

F

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