Short Answer

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

The given statement is false.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Statement of a Pythagorean Theorem.

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

Then the equation role="math" localid="1659507544082" $∥ \vec{x} + \vec{y} ∥^{2} = ∥ \vec{x} ∥^{2} + ∥ \vec{y} ∥^{2}$ 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 $\overset{⇀}{x} a n d \overset{⇀}{y}$ and are two vectors in $R^{n}$ , then the equation role="math" localid="1659507570743" $∥ \vec{x} + \vec{y} ∥^{2} = ∥ \vec{x} ∥^{2} + ∥ \vec{y} ∥^{2}$ must hold.

By the Pythagorean Theorem, the equation $∥ \vec{x} + \vec{y} ∥^{2} = ∥ \vec{x} ∥^{2} + ∥ \vec{y} ∥^{2}$ is holds only when $\overset{⇀}{x}$ and $\overset{⇀}{y}$ are orthogonal.

Then the given statement is false.

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Linear Algebra With Applications

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Short Answer

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

Step by Step Solution

Step 1: Statement of a Pythagorean Theorem.

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

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Linear Algebra With Applications

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TRUE OR FALSE? If x⇀and y⇀are two vectors in Rn, then the equation role="math" localid="1659506190737" ∥x→+y→∥2=∥x→∥2+∥y→∥2must hold.

Step by Step Solution

Step 1: Statement of a Pythagorean Theorem.

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

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