Q37E Consider a linear transformation... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q37E (page 55)

Consider a linear transformation Tfrom $R^{2}$ to $R^{2}$ . Suppose that $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ are two arbitrary vectors in $R^{2}$ and that $\overset{鈫�}{x}$ is a third vector whose endpoint is on the line segment connecting the endpoints of $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ . Is the endpoint of the vector $T (\overset{鈫�}{x)}$ necessarily on the line segment connecting the endpoints of $T (\overset{鈫�}{v)}$ and $T (\overset{鈫�}{w)}$ ? Justify your answer.

Short Answer

Expert verified

Thus, we can say that $T (\overset{鈫�}{x}) = T (\overset{鈫�}{v}) + k (T (\overset{鈫�}{w}) - T \overset{鈫�}{(v}))$ Is on line segment.

Step by step solution

Step by step Explanation Step1: Assuming the equation

We can write $\overset{鈫�}{x} = \overset{鈫�}{v} + k (\overset{鈫�}{w} - \overset{鈫�}{v})$ where k is any scalar between 0 and 1.

Linear Transformation

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

Identity of $R^{m}$ should be mapped to identity ofrole="math" localid="1659717675430" $R^{n}$ .
$T (a + b) = T (a) + T (b)$ For all $a, b 鈭� R^{m}$ .
$T (c a) = c T (a)$ Where c is any scalar and $a 鈭� R^{m}$ .

Solving the equations

On a similar basis we can write

$T (\overset{鈫�}{v} + k (\overset{鈫�}{w} - \overset{鈫�}{v}) = T (\overset{鈫�}{v}) + k T (\overset{鈫�}{w} - \overset{鈫�}{v}) T (\overset{鈫�}{v} + k (\overset{鈫�}{w} - \overset{鈫�}{v}) = T (\overset{鈫�}{v}) + k (T (\overset{鈫�}{w}) - T \overset{鈫�}{(v}))$

Thus, we can say that $T (\overset{鈫�}{x}) = T (\overset{鈫�}{v}) + k (T (\overset{鈫�}{w}) - T \overset{鈫�}{(v}))$ Is on line segment.

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91影视

Short Answer

Step by step solution

Step by step Explanation Step1: Assuming the equation

Linear Transformation

Solving the equations

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