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Q11E

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Found in: Page 263

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# TRUE OR FALSE? If $\stackrel{\mathbf{⇀}}{\mathbf{x}}{\mathbit{a}}{\mathbit{n}}{\mathbit{d}}{\mathbf{}}\stackrel{\mathbf{⇀}}{\mathbf{y}}$are two vectors in ${{\mathbit{R}}}^{{\mathbf{n}}}$, then the equation role="math" localid="1659506190737" $\mathbf{\parallel }\stackrel{\mathbf{\to }}{\mathbf{x}}\mathbf{+}\stackrel{\mathbf{\to }}{\mathbf{y}}{\mathbf{\parallel }}^{\mathbf{2}}\mathbf{=}\mathbf{\parallel }\stackrel{\mathbf{\to }}{\mathbf{x}}{\mathbf{\parallel }}^{\mathbf{2}}\mathbf{+}\mathbf{\parallel }\stackrel{\mathbf{\to }}{\mathbf{y}}{\mathbf{\parallel }}^{\mathbf{2}}$must hold.

The given statement is false.

See the step by step solution

## Step 1: Statement of a Pythagorean Theorem.

Suppose $\stackrel{\mathbf{⇀}}{\mathbf{x}}{\mathbit{a}}{\mathbit{n}}{\mathbit{d}}{\mathbf{}}\stackrel{\mathbf{⇀}}{\mathbf{y}}$and are two vectors in ${{\mathbit{R}}}^{{\mathbf{n}}}$ .

Then the equation role="math" localid="1659507544082" ${\mathbf{\parallel }}\stackrel{\mathbf{\to }}{\mathbf{x}}{\mathbf{+}}\stackrel{\mathbf{\to }}{\mathbf{y}}{{\mathbf{\parallel }}}^{{\mathbf{2}}}{\mathbf{=}}{\mathbf{\parallel }}\stackrel{\mathbf{\to }}{\mathbf{x}}{{\mathbf{\parallel }}}^{{\mathbf{2}}}{\mathbf{+}}{\mathbf{\parallel }}\stackrel{\mathbf{\to }}{\mathbf{y}}{{\mathbf{\parallel }}}^{{\mathbf{2}}}$ holds if and only if and are orthogonal.

## Step 2: Check whether the given statement is a true or false.

The given statement is if $\stackrel{⇀}{x}and\stackrel{⇀}{y}$ and are two vectors in ${R}^{n}$, then the equation role="math" localid="1659507570743" ${\parallel }\stackrel{\to }{\mathrm{x}}{+}\stackrel{\to }{\mathrm{y}}{{\parallel }}^{{2}}{=}{\parallel }\stackrel{\to }{\mathrm{x}}{{\parallel }}^{{2}}{+}{\parallel }\stackrel{\to }{\mathrm{y}}{{\parallel }}^{{2}}$must hold.

By the Pythagorean Theorem, the equation ${\parallel }\stackrel{\to }{\mathrm{x}}{+}\stackrel{\to }{\mathrm{y}}{{\parallel }}^{{2}}{=}{\parallel }\stackrel{\to }{\mathrm{x}}{{\parallel }}^{{2}}{+}{\parallel }\stackrel{\to }{\mathrm{y}}{{\parallel }}^{{2}}$ is holds only when $\stackrel{⇀}{x}$ and $\stackrel{⇀}{y}$are orthogonal.

Then the given statement is false.