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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 199)

There exist $2 脳 2$ matrix Asuch that the space Vof all matrices commuting with Ais one-dimensional.

Short Answer

Expert verified

The statement is False.

Step by step solution

01

Determine the set.

Consider a $2 脳 2$ matrix A from the space V that commute with all the element of where . $S = \{B 鈭� M^{2 脳 2} | AB = BA\}$

02

If A=0, determine the set S .

If $A = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ then for any $B 鈭� M^{2 脳 2}$ the matrix A commute with B implies $S = M^{2 脳 2}$ .

Therefore, the dimension of the set S is if $A = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ .

03

If A=1001, determine the set S .

If $A = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ then for any $B 鈭� M^{2 脳 2}$ the matrix A commute with B implies $S = M^{2 脳 2}$ .

Therefore, the dimension of the set S is 4 if $A = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ .

04

If A is linearly independent with J , determine the set S .

If A is linearly independent with respect to l then the set S contain at-least two elements A and l.

Therefore, the dimension of the set S is at least 2 if A is linearly independent with respect to l.

Hence, the statement is false.

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Most popular questions from this chapter

Find the image and kernel of the transformation $T$ in $T (x_{0}, x_{1}, x_{2}, x_{3}, . . .) = (0, x_{0}, x_{2}, x_{4})$ from $V$ to $V$ .

Show that in an n-dimensional linear space we can find at most n linearly independent elements.

In Exercise 72through 74, let $z_{n}$ be the set of all polynomials of degree $鈮� n$ such that $f (0) = 0$ .

72. Show that $z_{n}$ is a subspace of $p_{n}$ and find the dimension of $z_{n}$ .

Show that the space of infinite sequence of real numbers is infinite dimensional.

In Exercises 5 through 40, find the matrix of the given linear transformationwith respect to the given basis. If no basis is specified, use standard basis: $渭 = (1, t, t)$ for $P_{2}$ ,

role="math" localid="1659423247247" $渭 = ([\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}])$

for $^{2 脳 2}$ androle="math" localid="1659421462939" $渭 = (1, i)$ for,.For the space $U^{2 脳 2}$ of upper triangular $2 脳 2$ matrices, use the basis

$渭 = ([\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}])$

Unless another basis is given. In each case, determine whether Tis an isomorphism. If Tisn鈥檛 an isomorphism, find bases of the kernel and image of Tand thus determine the rank of T.

12. T $(M) = M [\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix}]$ from $R^{2 脳 2}$ to $R^{2 脳 2}$ .

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