§5.5, Exercise 4. Find a basis for the null space of10 -12A =1-14-51-2What is the dimension of the null space of A?

The given matrix is A=10-121-1004-51-2. First convert the above matrix into reduced row echelon form…

Answered: $5.5, Exercise 4. Find a basis for the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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§5.5, Exercise 4. Find a basis for the null space of
1
0 -1
2
A =
1
-1
4
-5
1
-2
What is the dimension of the null space of A?

Expert Solution

Step 1

The given matrix is $A = (\begin{matrix} 1 & 0 & - 1 & 2 \\ 1 & - 1 & 0 & 0 \\ 4 & - 5 & 1 & - 2 \end{matrix})$ .

First convert the above matrix into reduced row echelon form using elementary row transformations as follows.

$(\begin{matrix} 1 & 0 & - 1 & 2 \\ 1 & - 1 & 0 & 0 \\ 4 & - 5 & 1 & - 2 \end{matrix}) R_{2} \to R_{2} - R_{1}, R_{3} \to R_{3} - 4 R_{1} (\begin{matrix} 1 & 0 & - 1 & 2 \\ 0 & - 1 & 1 & - 2 \\ 0 & - 5 & 5 & - 10 \end{matrix}) R_{2} \to - R_{2} (\begin{matrix} 1 & 0 & - 1 & 2 \\ 0 & 1 & - 1 & 2 \\ 0 & - 5 & 5 & - 10 \end{matrix}) R_{3} \to R_{3} + 5 R_{2} (\begin{matrix} 1 & 0 & - 1 & 2 \\ 0 & 1 & - 1 & 2 \\ 0 & 0 & 0 & 0 \end{matrix})$

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