Q19E Give a geometric interpretation ... [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: Q19E (page 54)

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 $13$
$[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]$

Short Answer

Expert verified

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:

Step by step solution

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

Let the matrix be,

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

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

Compute the vectors.

Let the matrix be,

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

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

$\begin{array}{rcl} T (\overset{鈫�}{x}) & = & [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] \overset{鈫�}{x} \\ 鈬� & T [\begin{matrix} 1 \\ 0 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} 1 \\ 0 \end{matrix}] \\ 鈭� T [\begin{matrix} 1 \\ 0 \end{matrix}] & = & [\begin{matrix} 1 \\ 0 \end{matrix}] \end{array}$

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

$\begin{array}{rcl} T (\overset{鈫�}{x}) & = & [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] \overset{鈫�}{x} \\ 鈬� & T [\begin{matrix} 0 \\ 2 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} 0 \\ 2 \end{matrix}] \\ 鈭� T [\begin{matrix} 0 \\ 2 \end{matrix}] & = & [\begin{matrix} 0 \\ 0 \end{matrix}] \end{array}$

Graph the letter using matrix.

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

$T \overset{鈫�}{(x)}$ is obtained by rotating the vector $\overset{鈫�}{x}$ through an angle $90 掳$ of in the clockwise direction.

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 $a d - b c 鈮� 0$ $a d - b c 鈮� 0$ .

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,

$T = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] 鈬� (1 脳 0) - (0 脳 0) = 0$

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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91影视

Short Answer

Step by step solution

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

Compute the vectors.

Graph the letter using matrix.

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

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