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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q3.3-4E (page 143)

In Exercises 1through 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 $A$ .
$(\begin{matrix} 1 & 4 \\ 2 & 8 \end{matrix})$

Short Answer

Expert verified

The relation between the vectors is, ${\overset{鈫�}{v}}_{2} = 4 {\overset{鈫�}{v}}_{1}$ . The basis of the image is, $[\begin{matrix} 1 \\ 2 \end{matrix}]$ and the basis of the kernel is, $[\begin{matrix} 4 \\ - 1 \end{matrix}]$ .

Step by step solution

01

Consider the matrix

The matrix is,

$(\begin{matrix} 1 & 4 \\ 2 & 8 \end{matrix})$

02

Check for the image and kernel of the matrix

The vectors are ${\overset{鈫�}{v}}_{1} = [\begin{matrix} 1 \\ 2 \end{matrix}]$ and ${\overset{鈫�}{v}}_{2} = [\begin{matrix} 4 \\ 8 \end{matrix}]$ .

The relation can be defined as,

${\overset{鈫�}{v}}_{2} = 4 {\overset{鈫�}{v}}_{1} 4 {\overset{鈫�}{v}}_{1} - {\overset{鈫�}{v}}_{2} = 0 [\begin{matrix} {\overset{鈫�}{v}}_{1} \\ {\overset{鈫�}{v}}_{2} \end{matrix}] = [\begin{matrix} 4 \\ - 1 \end{matrix}]$

The image of the matrix is, $[\begin{matrix} 1 \\ 2 \end{matrix}]$ .

The kernel of the matrix is, $[\begin{matrix} 4 \\ - 1 \end{matrix}]$ .

03

Final answer

The relation between the vectors is, ${\overset{鈫�}{v}}_{2} = 4 {\overset{鈫�}{v}}_{1}$ . The basis of the image is, $[\begin{matrix} 1 \\ 2 \end{matrix}]$ and the basis of the kernel is, $[\begin{matrix} 4 \\ - 1 \end{matrix}]$ .

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Most popular questions from this chapter

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.

17. $[\begin{matrix} 0 & 1 & 2 & 0 & 3 \\ 0 & 0 & 0 & 1 & 4 \end{matrix}]$

Consider a nilpotent n 脳 n matrix A. Use the result demonstrated in exercise 78 to show that $A^{n} = 0$ .

What is the image of a function ffrom $鈩�$ to $鈩�$ given by

$f (t) = t^{3} + a t^{2} + b t + c$ ,

where a,b,c are arbitrary scalars?

For which value(s) of the constant k do the vectors below form a basis of ${鈩�}^{4}$ ?

$[\begin{matrix} 1 \\ 0 \\ 0 \\ 2 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \\ 3 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 1 \\ 4 \end{matrix}], [\begin{matrix} 2 \\ 3 \\ 4 \\ K \end{matrix}]$

Consider two n x m matrices A and B. What can you say about the relationship among the quantities rank(A), rank(B), rank(A+B).

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