Using the Gram–Schmidt orthonormalization process, construct the orthonormal set associated with the following set of vectors.\[\mathbf{e}_1 = [1 \, -1 \, 6 \, 1]^{\mathrm{T}}, \quad \mathbf{e}_2 = [-2 \, 1 \, 3 \, 2]^{\mathrm{T}}, \quad \mathbf{e}_3 = [3 \, 0 \, 4 \, 4]^{\mathrm{T}}, \quad \mathbf{e}_4 = [2 \, -1 \, 1 \, 0]^{\mathrm{T}}.\]

Answered: Using the Gram-Schmidt…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.5: Basis And Dimension

Problem 65E: Find a basis for the vector space of all 33 diagonal matrices. What is the dimension of this vector...

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Question

Using the Gram–Schmidt orthonormalization process, construct the orthonormal set associated with the following set of vectors.

\[
\mathbf{e}_1 = [1 \, -1 \, 6 \, 1]^{\mathrm{T}}, \quad \mathbf{e}_2 = [-2 \, 1 \, 3 \, 2]^{\mathrm{T}}, \quad \mathbf{e}_3 = [3 \, 0 \, 4 \, 4]^{\mathrm{T}}, \quad \mathbf{e}_4 = [2 \, -1 \, 1 \, 0]^{\mathrm{T}}.
\]

Expert Solution

Step 1

Given:

Vectors:

$e_{1} = {[1, - 1, 6, 1]}^{T}$

$e_{2} = {[- 2, 1, 3, 2]}^{T}$

$e_{3} = {[3, 0, 4, 4]}^{T}$

$e_{4} = {[2, - 1, 1, 0]}^{T}$

To find:

Orthonormal basis.

Gram Schmidt process:

Let $w_{1} = v_{1}$

Find $w_{2}$ as $w_{2} = v_{2} - \frac{<v_{2}, w_{1}>}{<w_{1}, w_{1}>} w_{1}$

Find $w_{3}$ as $w_{3} = v_{3} - \frac{<v_{3}, w_{1}>}{<w_{1}, w_{1}>} w_{1} - \frac{<v_{3}, w_{2}>}{<w_{2}, w_{2}>} w_{2}$

Find $w_{4}$ as $w_{4} = v_{4} - \frac{<v_{4}, w_{1}>}{<w_{1}, w_{1}>} w_{1} - \frac{<v_{4}, w_{2}>}{<w_{2}, w_{2}>} w_{2} - \frac{<v_{4}, w_{3}>}{<w_{3}, w_{3}>} w_{3}$

The set of vectors $\{w_{1}, w_{2}, w_{3}, w_{4}\}$ form an orthogonal basis.

The set of vectors $\{\frac{w_{1}}{||w_{1}||}, \frac{w_{2}}{||w_{2}||}, \frac{w_{3}}{||w_{3}||}, \frac{w_{4}}{||w_{4}||}\}$ form an orthonormal basis.