Short Answer

Consider the orthonormal vectors ${\vec{u}}_{1}, {\vec{u}}_{2}, {\vec{u}}_{3}, {\vec{u}}_{4}, {\vec{u}}_{5}$ in $ℝ^{10}$ . Find the length of the vector $\vec{x} = 7 {\vec{u}}_{1} - 3 {\vec{u}}_{2} + 2 {\vec{u}}_{3} + {\vec{u}}_{4} - {\vec{u}}_{5}$ .

The length of the vector $\vec{x} = 7 {\vec{u}}_{1} - 3 {\vec{u}}_{2} + 2 {\vec{u}}_{3} + {\vec{u}}_{4} - {\vec{u}}_{5}$ is $8$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step by Step Solution Step 1: Properties

Consider an orthonormal vector ${\vec{u}}_{1}, {\vec{u}}_{2}, {\vec{u}}_{3}, {\vec{u}}_{4}, {\vec{u}}_{5}$ in $ℝ^{10}$ such that $\vec{x} = 7 {\vec{u}}_{1} - 3 {\vec{u}}_{2} + 2 {\vec{u}}_{3} + {\vec{u}}_{4} - {\vec{u}}_{5}$ .

The property of dot product is defined as follows:

$(\vec{x} + \vec{y}) \cdot \vec{w} = \vec{x} \cdot \vec{w} + \vec{y} \cdot \vec{w}$
$(c \vec{x}) \cdot \vec{w} = c (\vec{x} \cdot \vec{w})$
${\vec{u}}_{i} \cdot {\vec{u}}_{j} = \{\begin{matrix} 0 & i f i \neq j \\ 1 & i f i = j \end{matrix}\}$

Step 2: Determine the length of the vector x→

Simplify the equation ${||\vec{x}||}^{2} = \vec{x} \cdot \vec{x}$ as follows:

role="math" localid="1659539563887" ${||\vec{x}||}^{2} = \vec{x} \cdot \vec{x} {||\vec{x}||}^{2} = \{7 {\vec{u}}_{1} - 3 {\vec{u}}_{2} + 2 {\vec{u}}_{3} + {\vec{u}}_{4} - {\vec{u}}_{5}\} \cdot \{7 {\vec{u}}_{1} - 3 {\vec{u}}_{2} + 2 {\vec{u}}_{3} + {\vec{u}}_{4} - {\vec{u}}_{5}\} {||\vec{x}||}^{2} = \{49 ({\vec{u}}_{1} \cdot {\vec{u}}_{1}) - 21 ({\vec{u}}_{1} \cdot {\vec{u}}_{2}) + 14 ({\vec{u}}_{1} \cdot {\vec{u}}_{3}) + 7 ({\vec{u}}_{1} \cdot {\vec{u}}_{4}) - 7 ({\vec{u}}_{1} \cdot {\vec{u}}_{5}) - 21 ({\vec{u}}_{2} \cdot {\vec{u}}_{1}) + 9 ({\vec{u}}_{2} \cdot {\vec{u}}_{2}) - 6 ({\vec{u}}_{2} \cdot {\vec{u}}_{3}) - 3 ({\vec{u}}_{2} \cdot {\vec{u}}_{4}) + 3 ({\vec{u}}_{2} \cdot {\vec{u}}_{5}) + 14 ({\vec{u}}_{3} \cdot {\vec{u}}_{1}) - 6 ({\vec{u}}_{3} \cdot {\vec{u}}_{2}) + 4 ({\vec{u}}_{3} \cdot {\vec{u}}_{3}) + 2 ({\vec{u}}_{3} \cdot {\vec{u}}_{4}) - 2 ({\vec{u}}_{3} \cdot {\vec{u}}_{5}) + 7 ({\vec{u}}_{4} \cdot {\vec{u}}_{1}) - 3 ({\vec{u}}_{4} \cdot {\vec{u}}_{2}) + 2 ({\vec{u}}_{4} \cdot {\vec{u}}_{3}) + ({\vec{u}}_{4} \cdot {\vec{u}}_{4}) - ({\vec{u}}_{4} \cdot {\vec{u}}_{5}) - 7 ({\vec{u}}_{5} \cdot {\vec{u}}_{1}) + 3 ({\vec{u}}_{5} \cdot {\vec{u}}_{2}) - 2 ({\vec{u}}_{5} \cdot {\vec{u}}_{3}) - ({\vec{u}}_{5} \cdot {\vec{u}}_{4}) + ({\vec{u}}_{5} \cdot {\vec{u}}_{5})\}$

By the property of dot product, simplify the equation as follows:

${||\vec{x}||}^{2} = 49 + 9 + 4 + 1 + 1 {||\vec{x}||}^{2} = 64 ||\vec{x}|| = 8$

Hence, the length of the vector $\vec{x} = 7 {\vec{u}}_{1} - 3 {\vec{u}}_{2} + 2 {\vec{u}}_{3} + {\vec{u}}_{4} - {\vec{u}}_{5}$ is $8$ .

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Short Answer

Consider the orthonormal vectors ${\vec{u}}_{1}, {\vec{u}}_{2}, {\vec{u}}_{3}, {\vec{u}}_{4}, {\vec{u}}_{5}$ in $ℝ^{10}$ . Find the length of the vector $\vec{x} = 7 {\vec{u}}_{1} - 3 {\vec{u}}_{2} + 2 {\vec{u}}_{3} + {\vec{u}}_{4} - {\vec{u}}_{5}$ .

Step by Step Solution

Step by Step Solution Step 1: Properties

Step 2: Determine the length of the vector x→

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Linear Algebra With Applications

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Consider the orthonormal vectors u→1,u→2,u→3,u→4,u→5 in ℝ10 . Find the length of the vector x→=7u→1-3u→2+2u→3+u→4-u→5 .

Step by Step Solution

Step by Step Solution Step 1: Properties

Step 2: Determine the length of the vector x→

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Consider the orthonormal vectors ${\vec{u}}_{1}, {\vec{u}}_{2}, {\vec{u}}_{3}, {\vec{u}}_{4}, {\vec{u}}_{5}$ in $ℝ^{10}$ . Find the length of the vector $\vec{x} = 7 {\vec{u}}_{1} - 3 {\vec{u}}_{2} + 2 {\vec{u}}_{3} + {\vec{u}}_{4} - {\vec{u}}_{5}$ .