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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: Q62E (page 201)

There exists a $3 脳 3$ matrix P such that the linear transformation $T (M) = M P - P M$ from $R^{3 脳 3}$ to $R^{3 脳 3}$ is an isomorphism.

Short Answer

Expert verified

The given statement is True

Step by step solution

01

Determine the isomorphism

Consider the linear transformation as follows $T : V 鈫� W$ is anis omorphism if kernel of T is zero.

02

Determine if the statement is True or False

Simplify as:

$T (M) = M P - P M \ker (T) = \{M 鈭� R^{3 脳 3} : T (M) = 0\} \ker (T) = \{M 鈭� R^{3 脳 3} : M P - P M = 0\} \ker (T) = \{M 鈭� R^{3 脳 3} : M P = P M\}$

Simplify further as follows.

$\ker (T) = \{0\}$

Therefore, the linear transformation $T (M) = M P - P M$ from $R^{3 脳 3}$ to $R^{3 脳 3}$ is an isomorphism.

Thus, the given statement 鈥渢here exists a $3 脳 3$ matrix P such that the linear transformation $T (M) = M P - P M$ from $R^{3 脳 3}$ to $R^{3 脳 3}$ is an isomorphism is true.

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Most popular questions from this chapter

Question: If T is a linear transformation from $P_{6} to P_{6}$ that transform $t^{k}$ into a polynomial of degree $k (f o r k = 0, 1, . . ., 6)$ (for ) then T must be an isomorphism鈥�.

Find the transformation is linear and determine whether they are isomorphism .

Find the image, kernel, rank, and nullity of the transformation $T$ in $T (f (t)) = f (7)$ from $P_{2}$ to $R$ .

State true or false, the space $R^{2 脳 3}$ is five-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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