Q22E Find the basis of all nxn diagon... [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: Q22E (page 176)

Find the basis of all nxn diagonal matrix, and determine its dimension.

Short Answer

Expert verified

The dimension of a basis of A is n which is spanned by

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

Step by step solution

Determine the matrix.

Consider the matrix $A = [\begin{matrix} a_{1} & 0 & 0 & 0 \\ 0 & a_{n} & 鈥� & 0 \\ 0 & 鈰� & 鈰� & 鈰� \\ 0 & 0 & 鈥� & a_{n} \end{matrix}]$ where $a_{i j}$ are real.

Simplify the equation $A = [\begin{matrix} a_{1} & 0 & 0 & 0 \\ 0 & a_{n} & 鈥� & 0 \\ 0 & 鈰� & 鈰� & 鈰� \\ 0 & 0 & 鈥� & a_{n} \end{matrix}]$ as follows.

$A = [\begin{matrix} a_{1} & 0 & 0 & 0 \\ 0 & a_{n} & 鈥� & 0 \\ 0 & 鈰� & 鈰� & 鈰� \\ 0 & 0 & 鈥� & a_{n} \end{matrix}] A = [\begin{matrix} a_{1} & 0 & 0 & 0 \\ 0 & a_{n} & 鈥� & 0 \\ 0 & 鈰� & 鈰� & 鈰� \\ 0 & 0 & 鈥� & a_{n} \end{matrix}] + A = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & a_{2} & 鈥� & 0 \\ 0 & 鈰� & 鈰� & 鈰� \\ 0 & 0 & 鈥� & 0 \end{matrix}] + . . . + A = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 鈥� & 0 \\ 0 & 鈰� & 鈰� & 鈰� \\ 0 & 0 & 鈥� & a_{n} \end{matrix}] A = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 鈥� & 0 \\ 0 & 鈰� & 鈰� & 鈰� \\ 0 & 0 & 鈥� & 0 \end{matrix}] + a_{2} [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 鈥� & 0 \\ 0 & 鈰� & 鈰� & 鈰� \\ 0 & 0 & 鈥� & 0 \end{matrix}] + . . . + a_{n} [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 鈥� & 0 \\ 0 & 鈰� & 鈰� & 鈰� \\ 0 & 0 & 鈥� & 1 \end{matrix}]$

Find the basis of each required space

$in the matrix, [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 鈥� & 0 \\ 0 & 鈰� & 鈰� & 鈰� \\ 0 & 0 & 鈥� & 0 \end{matrix}], [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 鈥� & 0 \\ 0 & 鈰� & 鈰� & 鈰� \\ 0 & 0 & 鈥� & 0 \end{matrix}], . . ., [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 鈥� & 0 \\ 0 & 鈰� & 鈰� & 鈰� \\ 0 & 0 & 鈥� & 1 \end{matrix}] are linear independent$

Therefore, the matrix A is spanned by .

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

Hence, the dimension of A is n and spanned by .

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91影视

Find the basis of all nxn diagonal matrix, and determine its dimension.

Short Answer

Step by step solution

Determine the matrix.

Find the basis of each required space

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