Q8E For each matrixA聽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: Q8E (page 119)

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

Short Answer

Expert verified

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

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 $A$ :

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

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

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

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

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91影视

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

Short Answer

Step by step solution

The kernel of a matrix

Find kernel of the given matrix

One App. One Place for Learning.

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91影视

For each matrixAin exercises 1 through 13, find vectors that span the kernel of A. Use paper and pencil.8.A=[111111111]A=[111111111]

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

Discrete Mathematics

Pure Maths

Statistics

Decision Maths

Applied Mathematics

Geometry

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.
8. $A = [\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}]$ $A = [\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}]$