Q3.1-13E For each matrix in exercises 1 t... [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: Q3.1-13E (page 119)

For each matrix in exercises 1 through 13, find vectors that span the kernel of . Use paper and pencil.
13. $A = [\begin{matrix} 1 & 2 & 0 & 0 & 3 & 0 \\ 0 & 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$

Short Answer

Expert verified

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

Step by step solution

Determine 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 $A$ :

role="math" localid="1664169265437" $A = [\begin{matrix} 1 & 2 & 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 1 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$

The above equation gives $x_{1} + 2 x_{2} + 3 x_{5} = 0$ , $x_{3} + 2 x_{5} = 0$ and $x_{4} + x_{5} = 0$ . This implies $x_{1} = - 2 x_{2} - 3 x_{5}$ , $x_{3} = - 2 x_{5}$ and $x_{4} = - x_{5}$ .

From the above calculation, we can say that the solution set of the linear system is $[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \end{matrix}] = [\begin{matrix} - 2 x_{2} - 3 x_{5} \\ x_{2} \\ - 2 x_{5} \\ - x_{5} \\ x_{5} \\ x_{6} \end{matrix}]$ . So, we have $\overset{鈫�}{x} = x_{2} [\begin{matrix} - 2 \\ 1 \\ 0 \\ 0 \\ 0 \\ x_{6} \end{matrix}] + x_{5} [\begin{matrix} - 3 \\ 0 \\ - 2 \\ - 1 \\ 1 \\ 0 \end{matrix}] + x_{6} [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{matrix}]$

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

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91影视

For each matrix in exercises 1 through 13, find vectors that span the kernel of . Use paper and pencil.
13. $A = [\begin{matrix} 1 & 2 & 0 & 0 & 3 & 0 \\ 0 & 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$

Short Answer

Step by step solution

Determine the kernel of a matrix

Find kernel of the given matrix

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91影视

For each matrix in exercises 1 through 13, find vectors that span the kernel of . Use paper and pencil.13.A=[120030001020000110000000000000]

Short Answer

Step by step solution

Determine 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

Calculus

Decision Maths

Statistics

Logic and Functions

Probability and Statistics

Study anywhere. Anytime. Across all devices.

For each matrix in exercises 1 through 13, find vectors that span the kernel of . Use paper and pencil.
13. $A = [\begin{matrix} 1 & 2 & 0 & 0 & 3 & 0 \\ 0 & 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$