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Linear Algebra

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Weighted Dot Product: (2,1)-Inner Product in $R^2$

LINALG-XPFKKV

Define an inner product on $\mathbb{R}^2$ by the rule if $\boldsymbol{u}=\begin{bmatrix} u_1 \\\ u_2 \end{bmatrix},\ \boldsymbol{v}=\begin{bmatrix} v_1 \\\ v_2 \end{bmatrix} \in \mathbb{R}^2$ then:

$$\langle \boldsymbol{u},\ \boldsymbol{v}\rangle = 2u_1v_1 + u_2v_2$$

Compute $\langle \boldsymbol{u},\ \boldsymbol{v}\rangle$ if $\boldsymbol{u}=\begin{bmatrix} 1 \\\ 2 \end{bmatrix}$ and $\boldsymbol{v}=\begin{bmatrix} -3 \\\ -1 \end{bmatrix}$.

A

$0$

B

$-5$

C

$-12$

D

$12$

E

$-6$

F

$-8$