Q16E Question: In Exercises 1 through... [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 3: Q16E (page 143)

Question: In Exercises 1 through 20, find the redundant column vectors of the given matrix A 鈥渂y inspection.鈥� Then find a basis of the image of A and a basis of the kernel of A.
16. $[\begin{matrix} 1 & - 2 & 0 & - 1 & 0 \\ 0 & 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}]$

Short Answer

Expert verified

The redundant column vectors of matrix A $\overset{鈫�}{v_{2}} and \overset{鈫�}{v_{4}}$ .

The basis of the image of A = $\{[1 0 0], [0 1 0], [0 0 1]\}$

The basis of the kernel of A = $\{[2 1 0 0 0], [1 0 - 5 1 0]\}$

Step by step solution

Finding the redundant vectors

Let $\overset{鈫�}{v_{1}} = [1 0 0], \overset{鈫�}{v_{2}} = [- 2 0 0], \overset{鈫�}{v_{3}} = [0 1 0], \overset{鈫�}{v_{4}} = [- 1 5 0], \overset{鈫�}{v_{5}} = [0 0 1]$

Here we have $\overset{鈫�}{v_{2}} = 2 \overset{鈫�}{v_{1}} and \overset{鈫�}{v_{4}} = - \overset{鈫�}{v_{3}} + 5 \overset{鈫�}{v_{3}}$

$鈬� \overset{鈫�}{v_{2}}$ are $\overset{鈫�}{v_{4}}$ redundant vectors and $\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{3}} and \overset{鈫�}{v_{5}}$ are non-redundant vectors.

Finding the basis of the image of A

The non-redundant column vectors of A form the basis of the image of A.

Since column vectors $\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{3}} and \overset{鈫�}{v_{5}}$ non-redundant vectors of matrix A, thus the basis of the image A = $\{[1 0 0], [0 1 0], [0 0 1]\}$ i.e. .

Finding the basis of the kernel of A

$\overset{鈫�}{v_{2}} and \overset{鈫�}{v_{4}}$ Since the vectors are redundant vectors such that

$\overset{鈫�}{v_{2}} = 2 \overset{鈫�}{v_{1}} and \overset{鈫�}{v_{4}} = - \overset{鈫�}{v_{1}} + 5 \overset{鈫�}{v_{3}} 鈬� 2 \overset{鈫�}{v_{1}} + \overset{鈫�}{v_{2}} + 0 and \overset{鈫�}{v_{1}} - 5 \overset{鈫�}{v_{3}} + \overset{鈫�}{v_{4}} = 0$

Thus the vectors in kernel of A is $\overset{鈫�}{w_{1}} = [2 1 0 0 0], \overset{鈫�}{w_{2}} = [1 0 - 5 1 0]$ .

Final Answer

The redundant column vectors of matrix A are $\overset{鈫�}{v_{2}} and \overset{鈫�}{v_{4}}$ .

The basis of the image of A = $\{[1 0 0], [0 1 0], [0 0 1]\}$

The basis of the kernel of A = $\{[2 1 0 0 0], [1 0 - 5 1 0]\}$

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91影视

Question: In Exercises 1 through 20, find the redundant column vectors of the given matrix A 鈥渂y inspection.鈥� Then find a basis of the image of A and a basis of the kernel of A.
16. $[\begin{matrix} 1 & - 2 & 0 & - 1 & 0 \\ 0 & 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}]$

Short Answer

Step by step solution

Finding the redundant vectors

Finding the basis of the image of A

Finding the basis of the kernel of A

Final Answer

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91影视

Question: In Exercises 1 through 20, find the redundant column vectors of the given matrix A 鈥渂y inspection.鈥� Then find a basis of the image of A and a basis of the kernel of A.16.[1-20-100015000001]

Short Answer

Step by step solution

Finding the redundant vectors

Finding the basis of the image of A

Finding the basis of the kernel of A

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Logic and Functions

Probability and Statistics

Applied Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

Question: In Exercises 1 through 20, find the redundant column vectors of the given matrix A 鈥渂y inspection.鈥� Then find a basis of the image of A and a basis of the kernel of A.
16. $[\begin{matrix} 1 & - 2 & 0 & - 1 & 0 \\ 0 & 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}]$