Q21E True or False.The linear transfo... [FREE SOLUTION]

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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: Q21E (page 199)

True or False.
The linear transformation $T (M) = [\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}] M from R^{2 脳 2} to R^{2 脳 2} have rank 1 .$

Short Answer

Expert verified

The given statement is False.

Step by step solution

Determine the matrix B .

Consider the function $T (M) = [\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}] M from R^{2 脳 2} to R^{2 脳 2}$ .

Aslocalid="1659426855825" $\{[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]\}$ is the basis element of $R^{2 脳 2}$ , the matrixBis defined as follows.

$T \{[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]\} = [\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 3 & 0 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] + 3 [\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}]$

$T \{[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]\} = 1 . u_{1} + 0 . u_{2} + 3 . u_{3} + 0 . u_{4}$

The image of T at point $[\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}]$ is defined as follows.

$T \{[\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}]\} = [\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}] [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] = [\begin{matrix} 0 & 1 \\ 0 & 3 \end{matrix}] = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] + 3 [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}] = 0 . u_{1} + 1 . u_{2} + 0 . u_{3} + 3 . u_{4}$

The image of T at point $[\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}]$ is defined as follows.

$T \{[\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}]\} = [\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}] [\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}] = [\begin{matrix} 2 & 0 \\ 6 & 0 \end{matrix}] = 2 [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] + 6 [\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}] = 2 . u_{1} + 0 . u_{2} + 6 . u_{3} + 0 . u_{4}$

The image of T at point $[\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]$ is defined as follows.

$T \{[\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]\} = [\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}] [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 0 & 2 \\ 0 & 6 \end{matrix}] = 2 [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] + 6 [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}] T \{[\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]\} = 0 . u_{1} + 2 . u_{2} + 0 . u_{3} + 6 . u_{4}$

Therefore, the matrix B= $[\begin{matrix} 1 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 \\ 3 & 0 & 6 & 0 \\ 0 & 3 & 0 & 6 \end{matrix}]$ .

Determine the rank of T .

As two rows are linearly dependent implies rank (T)=2

Hence, the statement is false.

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91影视

True or False.
The linear transformation $T (M) = [\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}] M from R^{2 脳 2} to R^{2 脳 2} have rank 1 .$

Short Answer

Step by step solution

Determine the matrix B .

Determine the rank of T .

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91影视

True or False.The linear transformation T(M)=[1236]MfromR2脳2toR2脳2haverank1.

Short Answer

Step by step solution

Determine the matrix B .

Determine the rank of T .

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Most popular questions from this chapter

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True or False.
The linear transformation $T (M) = [\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}] M from R^{2 脳 2} to R^{2 脳 2} have rank 1 .$