Q3.1-15E For each matrix A聽in Exercise 1... [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: Q3.1-15E (page 119)

For each matrix $A$ in Exercise 14 through 16, find vectors that span the image of $A$ . Give as few vectors as possible. Use paper and pencil.
$A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \end{matrix}]$

Short Answer

Expert verified

Thus, the resulting vector that span the image of $A$ is $[\begin{matrix} 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \end{matrix}]$ .

Step by step solution

Span of the given matrix.

The given matrix is: $A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \end{matrix}]$ .

The objective is to obtain the vectors that span the image of A.

The column vectors of $A$ is solved as follows:

u_{1} = [\begin{matrix} 1 \\ 1 \end{matrix}], u_{2} = [\begin{matrix} 1 \\ 2 \end{matrix}], u_{3} = [\begin{matrix} 1 \\ 3 \end{matrix}], u_{4} = [\begin{matrix} 1 \\ 4 \end{matrix}]

The column vectors are span the image of A.

Possible set of vectors.

Let $x = [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}]$ then the equation solve as follows:

$T (x) = A x = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}]$

Now, apply row reduced form of the coefficient matrix $A$ is as follows:

$R_{2} 鈫� R_{2} - R_{1}, [\begin{matrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \end{matrix}] R_{1} 鈫� R_{1} - R_{2}, [\begin{matrix} 1 & 0 & - 1 & - 2 \\ 0 & 1 & 2 & 3 \end{matrix}]$

The matrix $A_{r r e f}$ has only 1^st and 2^nd column contain leading entries.

Therefore, the first and second columns of $A$ from the spanning set for the image of $A$ .

Thus, the set of vectors $\{u_{1}, u_{2}\}$ is the spanning set for the image of $A$ .

Therefore, the resulting vector that span the image of localid="1664174459782" $A$ is $[\begin{matrix} 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \end{matrix}]$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

For each matrix $A$ in Exercise 14 through 16, find vectors that span the image of $A$ . Give as few vectors as possible. Use paper and pencil.
$A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \end{matrix}]$

Short Answer

Step by step solution

Span of the given matrix.

Possible set of vectors.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Applied Mathematics

Decision Maths

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

91影视

For each matrix Ain Exercise 14 through 16, find vectors that span the image of A. Give as few vectors as possible. Use paper and pencil.A=[11111234]

Short Answer

Step by step solution

Span of the given matrix.

Possible set of vectors.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Applied Mathematics

Decision Maths

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

For each matrix $A$ in Exercise 14 through 16, find vectors that span the image of $A$ . Give as few vectors as possible. Use paper and pencil.
$A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \end{matrix}]$