Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Give a geometric interpretation of the linear transformations defined by the matrices in Exercises $16$ through $23$ . Show the effect of these transformations on the letter $L$ considered in Example $5$ . In each case, decide whether the transformation is invertible. Find the inverse if it exists, and interpret it geometrically. See Exercise $13$
$[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]$

The matrix $[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]$ has orthogonal projection onto the x-axis and is not invertible.

The geometrical interpretation is:

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step by Step Explanation: Step 1: Consider the matrix.

Let the matrix be,

$T (\vec{x}) = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] \vec{x}$

The letter L is made up of vectors $[\begin{matrix} 1 \\ 0 \end{matrix}]$ and $[\begin{matrix} 0 \\ 2 \end{matrix}]$

Step 2: Compute the vectors.

Let the matrix be,

$T (\vec{x}) = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] \vec{x}$

Consider the vector $[\begin{matrix} 1 \\ 0 \end{matrix}]$

$\begin{array}{rcl} T (\vec{x}) & = & [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] \vec{x} \\ \Rightarrow & T [\begin{matrix} 1 \\ 0 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} 1 \\ 0 \end{matrix}] \\ ∴ T [\begin{matrix} 1 \\ 0 \end{matrix}] & = & [\begin{matrix} 1 \\ 0 \end{matrix}] \end{array}$

Consider the vector $[\begin{matrix} 0 \\ 2 \end{matrix}]$

$\begin{array}{rcl} T (\vec{x}) & = & [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] \vec{x} \\ \Rightarrow & T [\begin{matrix} 0 \\ 2 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} 0 \\ 2 \end{matrix}] \\ ∴ T [\begin{matrix} 0 \\ 2 \end{matrix}] & = & [\begin{matrix} 0 \\ 0 \end{matrix}] \end{array}$

Step 3: Graph the letter using matrix.

Now, graph the original vectors and the obtained vectors as follow:

$T \vec{(x)}$ is obtained by rotating the vector $\vec{x}$ through an angle $90 °$ of in the clockwise direction.

Step 4: Check for the invertibility of the matrix and find the inverse if exists.

The matrix $[\begin{matrix} a & b \\ c & d \end{matrix}]$ is invertible if and only if $a d - b c \neq 0$ $a d - b c \neq 0$ .

The inverse of the matrix $[\begin{matrix} a & b \\ c & d \end{matrix}]$ is, ${[\begin{matrix} a & b \\ c & d \end{matrix}]}^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}]$ .

Consider the matrix,

$T = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] \Rightarrow (1 \times 0) - (0 \times 0) = 0$

Therefore, the given matrix is non-invertible and the shape of L gets transformed as a straight line along x-axis.

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Linear Algebra With Applications

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Short Answer

Step by Step Solution

Step by Step Explanation: Step 1: Consider the matrix.

Step 2: Compute the vectors.

Step 3: Graph the letter using matrix.

Step 4: Check for the invertibility of the matrix and find the inverse if exists.

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Linear Algebra With Applications

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Short Answer

Step by Step Solution

Step by Step Explanation: Step 1: Consider the matrix.

Step 2: Compute the vectors.

Step 3: Graph the letter using matrix.

Step 4: Check for the invertibility of the matrix and find the inverse if exists.

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