Mathematics

11.08.2022 20:16

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Does $v^\top a v = 0$ for all vectors v imply a is skew symmetric?

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alazayjaime1423

4,7(14 marks)

Let $\mathbf v=\mathbf x+\mathbf y$ . Then

$0=\left(\mathbf v^\top\mathbf{Av}\right)^\top$
$0=(\mathbf x+\mathbf y)^\top\mathbf A^\top(\mathbf x+\mathbf y)$
$0=\mathbf x^\top\mathbf A^\top\mathbf x+\mathbf x^\top\mathbf A^\top\mathbf y+\mathbf y^\top\mathbf A^\top\mathbf x+\mathbf y^\top\mathbf A^\top\mathbf y$
$0=\mathbf x^\top\mathbf A^\top\mathbf y+\mathbf y^\top\mathbf A^\top\mathbf x$
$\mathbf x^\top\mathbf A^\top\mathbf y=-\mathbf y^\top\mathbf A^\top\mathbf x$
$\mathbf x^\top\mathbf A^\top\mathbf y=-\left(\mathbf x^\top\mathbf{Ay}\right)^\top$

which requires $\mathbf A=-\mathbf A^\top$ , so the answer is yes.

brinks7994

4,4(87 marks)

x = 9

Step-by-step explanation:

7x + 3x - 18 = 6(x + 3)

10x - 18 = 6(x + 3) Add like terms

10x - 18 = 6x + 18 Distribute 6 to x and 3

4x - 18 = 18 Subtract 6x to both sides

4x = 36 Add 18 to both sides

x = 9 Divide 4 to both sides

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