Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.
8. $[\begin{matrix} 5 \\ 4 \\ 2 \\ 2 \end{matrix}], [\begin{matrix} 3 \\ 6 \\ 7 \\ - 2 \end{matrix}]$

The orthonormal vectors of the sequence $[\begin{matrix} 5 \\ 4 \\ 2 \\ 2 \end{matrix}], [\begin{matrix} 3 \\ 6 \\ 7 \\ - 2 \end{matrix}] i s [\begin{matrix} 5 / 7 \\ 4 / 7 \\ 2 / 7 \\ 2 / 7 \end{matrix}], [\begin{matrix} - 2 / 7 \\ 2 / 7 \\ 5 / 7 \\ - 4 / 7 \end{matrix}]$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Determine the Gram-Schmidt process

Consider a basis of a subspace V of $R^{n} f o r j = 2, . . . ., m$ we resolve the vector $v ⃗_{1}$ into its components parallel and perpendicular to the span of the preceding vectors $v ⃗_{1}, . . . . {, v}_{j - 1} ⃗$ ,

Then

$u ⃗_{1} = \frac{1}{| | v ⃗_{1} | |} v ⃗_{1}, u ⃗_{2} = \frac{1}{| | v {⃗_{2}}^{⊥} | |} v {⃗_{2}}^{⊥}, . . . . ., u ⃗_{j} = \frac{1}{| | v {⃗_{j}}^{⊥} | |} v {⃗_{j}}^{⊥}, . . . . ., u ⃗_{m} = \frac{1}{| | v {⃗_{m}}^{⊥} | |} v {⃗_{m}}^{⊥}$

Step 2: Apply the Gram-Schmidt process

Let the given vectors are $v ⃗_{1} = [\begin{matrix} 5 \\ 4 \\ 2 \\ 2 \end{matrix}], v ⃗_{2} = [\begin{matrix} 3 \\ 6 \\ 7 \\ - 2 \end{matrix}]$ .

Obtain the values of , according to the Gram-Schmidt process.

$u ⃗_{1} = \frac{v ⃗_{1}}{| | v ⃗ | |} . . . . (1) u ⃗_{2} = \frac{v ⃗_{2} - (u ⃗_{1} . v ⃗_{2}) u ⃗_{1}}{| | v ⃗_{2} - (u ⃗_{1} . v ⃗_{2} u ⃗_{1} | |} . . . . . (2)$

Find $u ⃗_{1}$ .

$u ⃗_{1} = \frac{1}{\sqrt{5^{2} + 4^{2} + 2^{2} + 2^{2}}} [\begin{matrix} 5 \\ 4 \\ 2 \\ 2 \end{matrix}] = \frac{1}{7} [\begin{matrix} 5 \\ 4 \\ 2 \\ 2 \end{matrix}]$

Step 3: Find u⃗2

Now, here is need to find out the values of $v ⃗_{2} - (u ⃗_{1} . v ⃗_{2}) u ⃗_{1} a n d | | v ⃗_{2} - (u ⃗_{1} . v ⃗_{2}) u ⃗_{1} | |$ to obtain the value of $u ⃗_{2}$ .

Consider the equations below.

role="math" localid="1659441637408" $\vec{u_{1}} \cdot \vec{v_{1}} = 7 (\vec{u_{1}} \cdot \vec{v_{1}}) \vec{u_{1}} = [\begin{matrix} 5 \\ 4 \\ 2 \\ 3 \end{matrix}] \vec{v_{1}} - (\vec{u_{1}} \cdot \vec{v_{1}}) \vec{u_{1}} = [\begin{matrix} 3 \\ 6 \\ 7 \\ - 2 \end{matrix}] - [\begin{matrix} 5 \\ 4 \\ 2 \\ 2 \end{matrix}] = [\begin{matrix} - 2 \\ 2 \\ 5 \\ - 4 \end{matrix}]$

Then,

$||\vec{v_{1}} - (\vec{u_{1}} \cdot \vec{v_{1}}) \vec{u_{1}}|| = \sqrt{4 + 4 + 25 + 16} = \sqrt{49} = 7$

Now find role="math" localid="1659441716297" $\vec{u_{2}}$ .

$\vec{u_{2}} = \frac{1}{7} [\begin{matrix} - 2 \\ 2 \\ 5 \\ - 4 \end{matrix}]$

Thus, the values of $u ⃗_{1}, u ⃗_{2}$ are $[\begin{matrix} 5 / 7 \\ 4 / 7 \\ 2 / 7 \\ 2 / 7 \end{matrix}], [\begin{matrix} - 2 / 7 \\ 2 / 7 \\ 5 / 7 \\ - 4 / 7 \end{matrix}]$ .

Hence, the orthonormal vectors are $[\begin{matrix} 5 / 7 \\ 4 / 7 \\ 2 / 7 \\ 2 / 7 \end{matrix}], [\begin{matrix} - 2 / 7 \\ 2 / 7 \\ 5 / 7 \\ - 4 / 7 \end{matrix}]$ .

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Linear Algebra With Applications

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Short Answer

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.
8. $[\begin{matrix} 5 \\ 4 \\ 2 \\ 2 \end{matrix}], [\begin{matrix} 3 \\ 6 \\ 7 \\ - 2 \end{matrix}]$

Step by Step Solution

Step 1: Determine the Gram-Schmidt process

Step 2: Apply the Gram-Schmidt process

Step 3: Find u⃗2

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Linear Algebra With Applications

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Short Answer

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.8. [5422],[367-2]

Step by Step Solution

Step 1: Determine the Gram-Schmidt process

Step 2: Apply the Gram-Schmidt process

Step 3: Find u⃗2

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Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.
8. $[\begin{matrix} 5 \\ 4 \\ 2 \\ 2 \end{matrix}], [\begin{matrix} 3 \\ 6 \\ 7 \\ - 2 \end{matrix}]$