Q45E Find a basis of the space V of a... [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: Q45E (page 177)

Find a basis of the space V of allmatrices A that commute with $B = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}]$ ,and thus determine the dimension of V.

Short Answer

Expert verified

The dimensions of the space V of all $3 脳 3$ matrices A is $\dim (V) = 3$ .

Step by step solution

Explanation for the dimension of the space.

If a linear space has a basis with n elements then all other bases of V, consists of n elements as well.

Also we say that n is the dimension of V

$d i m (V) = n$

Step 2:Solution for the dimension of the basis.

Consider, B is a diagonal with $3 脳 3$ matrices.

Assume $B = [\begin{array}{l} 1 鈥夆赌夆赌� 0 鈥夆赌夆赌� 0 \\ 0 鈥夆赌夆赌� 0 鈥夆赌夆赌夆€� 1 \\ 0 鈥夆赌夆赌� 0 鈥夆赌夆赌夆€� 0 \end{array}]$ is the $3 脳 3$ diagonal matrix of B

$\begin{array}{l} | 位 I 鈭� B | = 位 [\begin{array}{l} 1 鈥夆赌夆赌� 0 鈥夆赌夆赌� 0 \\ 0 鈥夆赌夆赌� 1 鈥夆赌夆赌� 0 \\ 0 鈥夆赌夆赌� 0 鈥夆赌夆赌� 1 \end{array}] 鈭� [\begin{array}{l} 1 鈥夆赌夆赌� 0 鈥夆赌夆赌� 0 \\ 0 鈥夆赌夆赌� 0 鈥夆赌夆赌� 1 \\ 0 鈥夆赌夆赌� 0 鈥夆赌夆赌� 0 \end{array}] \\ | 位 I 鈭� B | = 0 \\ [\begin{array}{l} 位鈥夆赌夆赌� 0 鈥夆赌夆赌� 0 \\ 0 鈥夆赌夆赌� 位鈥夆赌夆赌� 0 \\ 0 鈥夆赌夆赌� 0 鈥夆赌夆赌� 位鈥夆赌夆赌� \end{array}] 鈭� [\begin{array}{l} 1 鈥夆赌夆赌� 0 鈥夆赌夆赌� 0 \\ 0 鈥夆赌夆赌� 0 鈥夆赌夆赌� 1 \\ 0 鈥夆赌夆赌� 0 鈥夆赌夆赌� 0 \end{array}] = 0 \\ [\begin{array}{l} 位鈭� 1 鈥夆赌夆赌夆€夆赌� 0 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€� 0 \\ 0 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌� 位鈥夆赌夆赌夆€夆赌夆赌夆€夆赌� 鈭� 1 \\ 0 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌� 0 鈥夆赌夆赌夆€夆赌夆赌夆€夆赌夆赌� 位鈥夆赌夆赌� \end{array}] = 0 \end{array}$

Further simplify as follows

$\begin{array}{l} (位鈭� 1 鈥�) (位鈥�) (位鈥�) = 0 \\ 位 = 1, 0 鈥夆赌� o r 鈥� 0 \end{array}$

Thus the possible dimensions of the space V of all $3 脳 3$ matrices A that commute with B are $位 = 1, 0 鈥夆赌� o r 鈥� 0$ .

Step 3:Determination for the dimension of the space V.

Consider the matrix B which represents the basis about a space V in $R^{3}$

Here $B = [\begin{array}{l} 1 鈥夆赌夆赌� 0 鈥夆赌夆赌� 0 \\ 0 鈥夆赌夆赌� 0 鈥夆赌夆赌夆€� 1 \\ 0 鈥夆赌夆赌� 0 鈥夆赌夆赌夆€� 0 \end{array}]$ is the matrix of the basis about a space V

Here A is the $3 脳 3$ matrix of the basis about the space V.

Thus $n = 3$ in the matrices B of the basis about the space V.

$\begin{array}{l} \dim (V) = n \\ \dim (V) = 3 \end{array}$

Since the solution.

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91影视

Find a basis of the space V of allmatrices A that commute with $B = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}]$ ,and thus determine the dimension of V.

Short Answer

Step by step solution

Explanation for the dimension of the space.

Step 2:Solution for the dimension of the basis.

Step 3:Determination for the dimension of the space V.

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91影视

Find a basis of the space V of allmatrices A that commute withB=[010001000],and thus determine the dimension of V.

Short Answer

Step by step solution

Explanation for the dimension of the space.

Step 2:Solution for the dimension of the basis.

Step 3:Determination for the dimension of the space V.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Find a basis of the space V of allmatrices A that commute with $B = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}]$ ,and thus determine the dimension of V.