Q20E Find the basis of all matrices A... [FREE SOLUTION]

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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 4: Q20E (page 176)

Find the basis of all matrices $A = [\begin{matrix} a & b \\ c & d \end{matrix}] in R^{2 x 2}$ in such that a+d=0,and determine its dimension.

Short Answer

Expert verified

The dimension of a 2x2 matrix A such that a+d=0 is which is spanned by .

$S p a n \{[\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}], [\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]\} .$

Step by step solution

Determine the matrix.

Consider the matrix $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ such that a + d = 0are real.

Simplify the equation a + d = 0 as follows.

$a + d = 0 a = - d$

Substitute the value -a for d in the equation $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ as follows.

role="math" localid="1659413798349" $A = [\begin{matrix} a & b \\ c & d \end{matrix}] A = [\begin{matrix} a & b \\ c & - a \end{matrix}] A = [\begin{matrix} a & 0 \\ 0 & - a \end{matrix}] + [\begin{matrix} 0 & 0 \\ b & 0 \end{matrix}] + [\begin{matrix} 0 & 0 \\ 0 & c \end{matrix}] A = a [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] + b [\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}] + c [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]$

Find the basis of each required space

In the matrix, $[\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}], [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] and [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ and are linear independent.

Therefore, the matrix A is spanned byspan $\{[\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}], [\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]\} .$ .

Hence, the dimension of A is 3 and spanned by $s p a n \{[\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}], [\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]\} .$ .

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91影视

Find the basis of all matrices $A = [\begin{matrix} a & b \\ c & d \end{matrix}] in R^{2 x 2}$ in such that a+d=0,and determine its dimension.

Short Answer

Step by step solution

Determine the matrix.

Find the basis of each required space

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91影视

Find the basis of all matrices A=[abcd]inR2x2in such that a+d=0,and determine its dimension.

Short Answer

Step by step solution

Determine the matrix.

Find the basis of each required space

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Logic and Functions

Calculus

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Geometry

Study anywhere. Anytime. Across all devices.

Find the basis of all matrices $A = [\begin{matrix} a & b \\ c & d \end{matrix}] in R^{2 x 2}$ in such that a+d=0,and determine its dimension.