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Linear Algebra

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Determining Axioms of Inner Products: Negative Diagonal

LINALG-QJMKXD

Define a product $\langle\ ,\ \rangle : \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$ by the rule:

$$\langle \boldsymbol{u},\ \boldsymbol{v}\rangle = \left(\begin{bmatrix} 1 & 1 \\\ 0 & -2 \end{bmatrix}\boldsymbol{u}\right)\cdot\left(\begin{bmatrix} 1 & 1 \\\ 0 & -2 \end{bmatrix}\boldsymbol{v}\right)$$

Which axioms of an inner product does the above product satisfy?

Select ALL that apply.

A

$\langle \boldsymbol{u},\ \boldsymbol{v}\rangle = \langle \boldsymbol{v},\ \boldsymbol{u}\rangle$ for all $\boldsymbol{u}, \boldsymbol{v} \in \mathbb{R}^2$

B

$\langle c\boldsymbol{u},\ \boldsymbol{v}\rangle = \langle \boldsymbol{u},\ c\boldsymbol{v}\rangle = c\langle \boldsymbol{u},\ \boldsymbol{v}\rangle$ for all $\boldsymbol{u}, \boldsymbol{v} \in \mathbb{R}^2$ and $c\in \mathbb{R}$

C

$\langle \boldsymbol{u} + \boldsymbol{v},\ \boldsymbol{w}\rangle = \langle \boldsymbol{u},\ \boldsymbol{w}\rangle + \langle \boldsymbol{v},\ \boldsymbol{w}\rangle$ for all $\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w} \in \mathbb{R}^2$

D

$\langle \boldsymbol{u},\ \boldsymbol{u}\rangle \ge 0$ for all $\boldsymbol{u} \in \mathbb{R}^2$

E

$\langle \boldsymbol{u},\ \boldsymbol{u}\rangle = 0$ if and only if $\boldsymbol{u} = \boldsymbol{0}$

F

None of the above