Q16E Find the basis of each of the sp... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 4: Q16E (page 176)

Find the basis of each of the space $V = R^{3 脳 2}$ ,and determine its dimension.

Short Answer

Expert verified

The dimension of a $3 脳 2$ matrix is 6 which is spanned byrole="math" localid="1659416482326" $S p a n \{[\begin{matrix} 1 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 0 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 0 \\ 0 & 1 \end{matrix}]\}$ .

Step by step solution

Determine the matrix

Consider the matrix $A = [\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{matrix}]$ where a and b are real.

Simplify the equation $A = [\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{matrix}]$ as follows.

role="math" localid="1659416376950" $A = [\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{matrix}] A = [\begin{matrix} a_{11} & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix}] + [\begin{matrix} 0 & a_{12} \\ 0 & 0 \\ 0 & 0 \end{matrix}] + [\begin{matrix} 0 & 0 \\ a_{21} & 0 \\ 0 & 0 \end{matrix}] + a_{22} [\begin{matrix} 0 & 0 \\ 0 & a_{22} \\ 0 & 0 \end{matrix}] + [\begin{matrix} 0 & 0 \\ 0 & 0 \\ a_{31} & 0 \end{matrix}] + [\begin{matrix} 0 & 0 \\ 0 & 0 \\ 0 & a_{32} \end{matrix}] A = a_{11} [\begin{matrix} 1 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix}] + a_{12} [\begin{matrix} 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{matrix}] + a_{21} [\begin{matrix} 0 & 0 \\ 1 & 0 \\ 0 & 0 \end{matrix}] + a_{22} [\begin{matrix} 0 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix}] + a_{31} [\begin{matrix} 0 & 0 \\ 0 & 0 \\ 1 & 0 \end{matrix}] + a_{32} [\begin{matrix} 0 & 0 \\ 0 & 0 \\ 0 & 1 \end{matrix}]$

Find the basis of each required space

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

Therefore, the matrix A is spanned by $s p a n = \{[\begin{matrix} 1 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 0 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 0 \\ 0 & 1 \end{matrix}]\}$ .

Hence, the dimension of A is 6 and spanned by $s p a n = \{[\begin{matrix} 1 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 0 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 0 \\ 0 & 1 \end{matrix}]\}$ .

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91影视

Find the basis of each of the space $V = R^{3 脳 2}$ ,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 each of the spaceV=R3脳2,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

Pure Maths

Statistics

Probability and Statistics

Discrete Mathematics

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Find the basis of each of the space $V = R^{3 脳 2}$ ,and determine its dimension.