Q2E For each matrixA in exercises 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: Q2E (page 119)

For each matrix $A$ in exercises 1 through 13, find vectors that span the kernel of $A$ . Use paper and pencil.
2. $A = [\begin{matrix} 1 & 2 & 3 \end{matrix}]$

Short Answer

Expert verified

The kernel of $A$ is $\ker (A) = s p a n ([\begin{matrix} 鈭� 2 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 鈭� 3 \\ 0 \\ 1 \end{matrix}])$ .

Step by step solution

The kernel of a matrix

The kernel of a matrix $A$ is the solution set of the linear system $A \overset{鈫�}{x} = 0$ .

Find kernel of the given matrix

Solve the linear system $A \overset{鈫�}{x} = 0$ by reduced row-echelon form of $A$ :

$[\begin{matrix} 1 & 2 & 3 & 0 \end{matrix}]$

The above equation gives $x_{1} + 2 x_{2} + 3 x_{3} = 0$ . This implies $x_{1} = 鈭� 2 x_{2} 鈭� 3 x_{3}$ .

From the above calculation, we can say that the solution set of the linear system is $[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} 鈭� 2 x_{2} 鈭� 3 x_{3} \\ x_{2} \\ x_{3} \end{matrix}]$ . So, we have $\overset{鈫�}{x} = x_{2} [\begin{matrix} 鈭� 2 \\ 1 \\ 0 \end{matrix}] + x_{3} [\begin{matrix} 鈭� 3 \\ 0 \\ 1 \end{matrix}]$ .

Thus, $\ker (A) = s p a n ([\begin{matrix} 鈭� 2 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 鈭� 3 \\ 0 \\ 1 \end{matrix}])$ $\ker (A) = s p a n ([\begin{matrix} 鈭� 2 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 鈭� 3 \\ 0 \\ 1 \end{matrix}])$ .

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 exercises 1 through 13, find vectors that span the kernel of $A$ . Use paper and pencil.
2. $A = [\begin{matrix} 1 & 2 & 3 \end{matrix}]$

Short Answer

Step by step solution

The kernel of a matrix

Find kernel of the given matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Pure Maths

Probability and Statistics

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

For each matrixA in exercises 1 through 13, find vectors that span the kernel of A. Use paper and pencil.2.A=[123]

Short Answer

Step by step solution

The kernel of a matrix

Find kernel of the given matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Pure Maths

Probability and Statistics

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.

For each matrix $A$ in exercises 1 through 13, find vectors that span the kernel of $A$ . Use paper and pencil.
2. $A = [\begin{matrix} 1 & 2 & 3 \end{matrix}]$