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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: Q3.1-39E (page 120)

Consider an $鈥� n 脳 p$ matrix $鈥� A$ and $鈥� p 脳 m$ matrix $B$ .
What is the relationship between ker( $A B$ ) and ker role="math" localid="1664188035115" $B$ )? Are they always equal? Is one of them always contained in the other?
What is the relationship between im( $A$ ) and im (role="math" localid="1664187975738" $A B$ )?

Short Answer

Expert verified

$\ker (B) 鈯� \ker (A B)$
$i m (A B) 鈯� i m (A)$

Step by step solution

01

Define kernel of linear transformation

Thekernel of linear transformation is defined as follows:

The kernel of a linear transformation $T (\overset{鈫�}{x}) = A \overset{鈫�}{x}$ from ${鈩�}^{m}$ to ${鈩�}^{n}$ consists of all zeros of the transformation, i.e., the solutions of the equations $T (\overset{鈫�}{x}) = A \overset{鈫�}{x} = 0$ .

It is denoted by ker $(T)$ or ker $(A)$ .

Let $A$ be an $n 脳 p$ matrix and $B$ be an $p 脳 m$ matrix.

02

(a) Find relationship between kernels

Let $\overset{鈫�}{x} 鈭� \ker (B) .$

Therefore, the equation is:

$B \overset{鈫�}{x} = 0$ 鈥︹€� (1)

Now,

$(A B) \overset{鈫�}{x} = A (B \overset{鈫�}{x}) = A 路 0 = 0 鈥� 鈥� 鈥� 鈥� 鈰� 鈰� (鈭� 鈥� b y 鈥� (1))$

$鈭� (A B) \overset{鈫�}{x} = 0$

Thus, $\overset{鈫�}{x} 鈭� \ker (A B) .$

Therefore, $\ker (B) 鈯� \ker (A B)$ .

Hence, by Exercise 37, they do not need to be equal.

03

(b) Find relationship between images

Consider $\overset{鈫�}{z} 鈭� i m (A B)$ .

This implies that there exists a $\overset{鈫�}{y}$ such that

$(A B) \overset{鈫�}{y} = \overset{鈫�}{z} 鈬� A (B \overset{鈫�}{y}) = \overset{鈫�}{z} 鈬� \overset{鈫�}{z} 鈭� i m (A)$

Thus, $im (AB) 鈯� im (A)$ .

Hence, by Exercise 37, they do not need to be equal.

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Most popular questions from this chapter

We are told that a certain $5 脳 5$ matrix $A$ can be written as

$A = B C$ ,

where $B$ is $5 脳 4$ and $C$ is $4 脳 5$ . Explain how you know that $A$ is not invertible.

In Exercises 21 through 25, find the reduced row-echelon form of the given matrix A. Then find a basis of the image of A and a basis of the kernel of A.

24.

$24 路 [\begin{matrix} 4 & 8 & 1 & 1 & 6 \\ 3 & 6 & 1 & 2 & 5 \\ 2 & 4 & 1 & 9 & 10 \\ 1 & 2 & 3 & 2 & 0 \end{matrix}]$

Consider a nonzero vector $\overset{鈫�}{蠀}$ in ${鈩�}^{3}$ . Using a geometric argument, describe the kernel of the linear transformation $T$ from ${鈩�}^{3}$ to ${鈩�}^{3}$ given by,

$T (\overset{鈫�}{x}) = \overset{鈫�}{蠀} 脳 \overset{鈫�}{x}$

See Definition A.9 in the Appendix.

(a) Consider a linear transformation $T$ from ${鈩�}^{5}$ to ${鈩�}^{3}$ . What are the possible values of $d i m (k e r T)$ ? Explain.

(b) Consider a linear transformation $T$ from ${鈩�}^{4}$ to ${鈩�}^{7}$ . What are the possible values of $d i m (i m T)$ ? Explain.

Consider the plane $x_{1} + 2 x_{2} + x_{3} = 0$ . Find a basis of this plane such that ${[\overset{鈫�}{x}]}_{B} = [\begin{matrix} 2 \\ - 1 \end{matrix}] for \overset{鈫�}{x} = [\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}]$ .

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