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Linear Algebra

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Inner Product Space: Inner Product in $R^4$

LINALG-P4OVJL

Define an inner product on $\mathbb{R}^4$ by the rule if $\boldsymbol{u}=\begin{bmatrix} u_1 \\\ u_2 \\\ u_3 \\\ u_4 \end{bmatrix},\ \boldsymbol{v}=\begin{bmatrix} v_1 \\\ v_2 \\\ v_3 \\\ v_4 \end{bmatrix} \in \mathbb{R}^4$ then:

$$\langle \boldsymbol{u},\ \boldsymbol{v}\rangle = \dfrac{1}{2}u_1v_1 + 2u_2v_2 + 2u_3v_3 + \dfrac{1}{2}u_4v_4$$

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

A

$0$

B

$1$

C

$2$

D

$3$

E

$4$

F

$5$