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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: Q39E (page 144)

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.

Short Answer

Expert verified

Thus, it is proved matrix $A$ is not invertible.

Step by step solution

01

Given in the question.

Let matrix $A$ is $5 脳 5$ which is $A = B C$ where $B$ is $5 脳 4$ and $C$ is $4 脳 5$ .

Let assume that $c o l s p a c e (A) 鈯� c o l s p a c e (B)$ . There are different ways and one way is as the linear transformation $T_{A}$ is the composition of $T_{B}$ and $T_{C}$ gives $i m (T_{A}) 鈯� i m (T_{B})$ .

02

Explanation of A is not invertible.

Take dimensions where $r a n k (A) 鈮� r a n k (B)$ . Now by the rank-nullity theorem applied to $B$ , gives $r a n k (B) 鈮� 4$ .

Which implies that $r a n k (A)$ is never 5, that is needed for $A$ to be invertible.

Hence, matrix $A$ is not invertible.

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Most popular questions from this chapter

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

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

Explain why fitting a cubic through the mpoints $P_{1} (x_{1}, y_{1}), . . . . . ., P_{m} (x_{m}, y_{m})$ amounts to finding the kernel of an mx10matrix A. Give the entries of theof row A.

Consider a 5x4matrix $A = [\begin{matrix} | & | & | & | \\ {\overset{鈫�}{V}}_{1} & {\overset{鈫�}{V}}_{2} & {\overset{鈫�}{V}}_{3} & {\overset{鈫�}{V}}_{4} \\ | & | & | & | \end{matrix}]$ . We are told that the vector $[\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}]$ is in the kernel of A. Write ${\overset{鈫�}{v}}_{4}$ as a linear combination of ${\overset{鈫�}{V}}_{1}, {\overset{鈫�}{V}}_{2}, {\overset{鈫�}{V}}_{3}$ .

In the accompanying figure, sketch the vector $\overset{鈫�}{x}$ with ${[\overset{鈫�}{x}]}_{貜} = [\begin{matrix} - 1 \\ 2 \end{matrix}]$ , where $貜$ is the basis of ${鈩�}^{2}$ consisting of the vectors $\overset{鈫�}{v}, \overset{鈫�}{w}$ .

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.

22. $[\begin{matrix} 2 & 4 & 8 \\ 4 & 5 & 1 \\ 7 & 9 & 3 \end{matrix}]$

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