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

Use the concept of a linear transformation in terms of the formula $y = A \overset{⇀}{x}$ , and interpret simple linear transformations geometrically. Find the inverse of a linear transformation from localid="1659964769815" $ℝ^{2} t o ℝ^{2}$ to (if it exists). Find the matrix of a linear transformation column by column.

Consider the transformations from $ℝ^{3} t o ℝ^{3}$ defined in Exercises 1 through 3. Which of these transformations are linear?
$y_{1} = 2 x_{2} y_{2} = x_{2} + 2 y_{3} = 2 x_{2}$

The given transformation $y_{1} = 2 x_{2} y_{2} = x_{2} + 2 y_{3} = 2 x_{2}$ is not linear.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: System of equations

We have given that system of equation is . $y_{1} = 2 x_{2} y_{2} = x_{2} + 2 y_{3} = 2 x_{2}$

For the transformation of $R^{3} t o R^{3}$ it can be written as:

$T (x_{1}, x_{2}, x_{3}) = (y_{1}, y_{2}, y_{3}) T (x_{1}, x_{2}, x_{3}) = (2 x_{2}, x_{2} + 2, 2 x_{2})$

Step2: Linear Transformation

A transformation from $R^{m} \to R^{n}$ is said to be linear if the following condition holds:

Identity of R^m should be mapped to identity of Rⁿ .
For all $T (a + b) = T (a) + T (b)$ .

3. $T (c a) = c T (a)$ Where c is any scalar and $a \in R^{m}$

Step 3: Checking for linear transformation

Identity of R³ is (0,0,0).

Now we will find the transformation of identity element.

$T (0, 0, 0) = (2 (0), 0 + 2, 2 (0)) = (0, 2, 0) \neq (0, 0, 0)$

First condition of linear transformation does not hold.

Thus, it is not a linear transformation.

Find Study Materials for

Create Study Materials

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Step by Step Solution

Step1: System of equations

Step2: Linear Transformation

Step 3: Checking for linear transformation

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

Step by Step Solution

Step1: System of equations

Step2: Linear Transformation

Step 3: Checking for linear transformation

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