Determine whether th e set ẞ is a basis for the vector space V V = M22»1 22 11 32 3[1 2B =21 2-3 1]3 13 2

Answered: V = M22» 1 2 2 1 1 3 2 3 [1 2 B = 2 1 2…

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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Question

Determine whether th e set ẞ is a basis for the vector space V

V = M22»
1 2
2 1
1 3
2 3
[1 2
B =
2
1 2
-3 1]
3 1
3 2

Expert Solution

Step 1

Basis: A set S is said to be a basis of a vector space if it is linearly independent and S spans the vector

space V.

We have to determine the basis of V. For this, we have to check the linearly independent matrices in

the set B.

Step 2

We can always write the entries of the matrix $[\begin{matrix} a & b \\ c & d \end{matrix}]$ as a vector $(a, b, c, d)$ . So we get,

$\begin{array}{rcl} [\begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}] & \to & (1, 2, 2, 1) \\ [\begin{matrix} 2 & 1 \\ 1 & 2 \end{matrix}] & \to & (2, 1, 1, 2) \\ [\begin{matrix} 1 & 3 \\ - 3 & 1 \end{matrix}] & \to & (1, 3, - 3, 1) \\ [\begin{matrix} 2 & 3 \\ 3 & 1 \end{matrix}] & \to & (2, 3, 3, 1) \\ [\begin{matrix} 1 & 2 \\ 3 & 2 \end{matrix}] & \to & (1, 2, 3, 2) \end{array}$

Writing these vectors as columns of a matrix, we get, $[\begin{matrix} 1 & 2 & 1 & 2 & 1 \\ 2 & 1 & 3 & 3 & 2 \\ 2 & 1 & - 3 & 3 & 3 \\ 1 & 2 & 1 & 1 & 2 \end{matrix}]$ .

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