Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

Answers without the blur.

Just sign up for free and you're in.

Short Answer

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

The given statement is False.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Determine the matrix B .

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

As localid="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 $R^{2 \times 2}$ , the matrix B is 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= $[\begin{matrix} 1 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 \\ 3 & 0 & 6 & 0 \\ 0 & 3 & 0 & 6 \end{matrix}]$ .

Step 2: Determine the rank of T .

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

Hence, the statement is false.

Find Study Materials for

Create Study Materials

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

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

Step by Step Solution

Step 1: Determine the matrix B .

Step 2: Determine the rank of T .

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

True or False.The linear transformation T(M)=[1236]M from R2×2 to R2×2 have rank 1.

Step by Step Solution

Step 1: Determine the matrix B .

Step 2: Determine the rank of T .

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

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