Q1E Using paper and pencil, perform ... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q1E (page 224)

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in exercises 1 through 14.
$1 . [\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}]$

Short Answer

Expert verified

The orthonormal vectors of the sequence $[\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}] i s \frac{1}{3} [\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}] .$

Step by step solution

Determine the Gram-Schmidt process.

Consider a basis of a subspace Vof $R^{n}$ for j=2,...., m we resolve the vector $\overset{鈬赌}{v_{j}}$ into its components parallel and perpendicular to the span of the preceding vectors $\overset{鈬赌}{v_{1}}, . . . . . . {\overset{鈬赌}{v}}_{j - 1}$ ,

Then

${\overset{鈬赌}{u}}_{1} = \frac{1}{||{\overset{鈬赌}{v}}_{1}||} {\overset{鈬赌}{v}}_{1}, {\overset{鈬赌}{u}}_{2} = \frac{1}{||{\overset{鈬赌}{v}}_{2}^{鈯�}||} {\overset{鈬赌}{v}}_{2}^{鈯�}, . . . . ., {\overset{鈬赌}{u}}_{j} = \frac{1}{||{\overset{鈬赌}{v}}_{j}^{鈯�}||} {\overset{鈬赌}{v}}_{j}^{鈯�}, . . . . ., {\overset{鈬赌}{u}}_{m} = \frac{1}{||{\overset{鈬赌}{v}}_{m}^{鈯�}||} {\overset{鈬赌}{v}}_{m}^{鈯�}$

According to Gram-Schmidt process

The given vector is $[\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}]$ .

Consider the value of ${\overset{鈬赌}{u}}_{1}$ according to Gram-Schmidt process.

${\overset{鈬赌}{u}}_{1} = \frac{1}{||{\overset{鈬赌}{v}}_{1}||} {\overset{鈬赌}{v}}_{1} . . . . . (1)$

Here, put the values in equation (1). We obtained

${\overset{鈬赌}{u}}_{1} = \frac{1}{\sqrt{2^{2} + 1^{2} + {(- 2)}^{2}}} . [\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}] = \frac{1}{3} . [\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}]$

Hence, the orthonormal vector is $\frac{1}{3} . [\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}]$ .

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91影视

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in exercises 1 through 14.
$1 . [\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}]$

Short Answer

Step by step solution

Determine the Gram-Schmidt process.

According to Gram-Schmidt process

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91影视

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in exercises 1 through 14.1.[21-2]

Short Answer

Step by step solution

Determine the Gram-Schmidt process.

According to Gram-Schmidt process

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Probability and Statistics

Applied Mathematics

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in exercises 1 through 14.
$1 . [\begin{matrix} 2 \\ 1 \\ - 2 \end{matrix}]$