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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: Q56E (page 200)

If the matrix of a linear transformation T ( with respect to some basis) is $[\begin{matrix} 3 & 5 \\ 0 & 4 \end{matrix}]$ then there must exists a non-zero element f in the domain of T such that T(f)=4f.

Short Answer

Expert verified

The given statement is True.

Step by step solution

01

Determine the concept for the basis

Consider the transformation as follows.

$T : V 鈫� V$ with respect to some basis B is as follows.

$B = [\begin{matrix} 3 & 5 \\ 0 & 4 \end{matrix}]$

Since B is a $2 脳 2$ matrix and V is a space of dimension 2.

Let the basis as follows.

$B = {v_{1}, v_{2}}$

02

Determine if the statement is true or false

Consider as follows $f = 5 W_{1} + V_{2}$ and as follows.

${[f]}_{a} = [\begin{matrix} 5 \\ 1 \end{matrix}]$ and then ${[T (v)]}_{B} = B {[v]}_{B}$ for v in V and as follows.

$\begin{array}{rcl} {[T (f)]}_{B} & = & [\begin{matrix} 3 & 5 \\ 0 & 4 \end{matrix}] [\begin{matrix} 5 \\ 1 \end{matrix}] \\ = & [\begin{matrix} 20 \\ 4 \end{matrix}] \\ = & 4 [\begin{matrix} 5 \\ 1 \end{matrix}] \\ = & 4 {[f]}_{B} \end{array}$

Solve further as:

$T (f) = 4 f$

Thus, 鈥渋f the matrix representation T is $[\begin{matrix} 3 & 5 \\ 0 & 4 \end{matrix}]$ , then there exists a non-zero element f in the domain of T such that $T (f) = 4 f$ is true.

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Most popular questions from this chapter

T denotes the space of infinity sequence of real numbers, $T (f (t)) = [\begin{matrix} f (5) \\ f (7) \\ f (11) \end{matrix}] f r o m P_{2} t o R^{3} .$ .

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 $R^{2 脳 2}$ .

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, $T (f (t)) = f (7)$ from $P_{2}$ torole="math" localid="1659412169328" $鈩�$ .

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

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

for ${鈩�}^{2 脳 2}$ and $渭 = (1, i)$ for, ${鈩�}^{2 脳 2}$ .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 $T$ is an isomorphism. If $T$ isn鈥檛 an isomorphism, find bases of the kernel and image of $T$ and thus determine the rank of $T$ .

17. $T (z) = iz$ from $C$ to $C$ with respect to the basis $B = {1, i}$ .

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 $V$ .

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