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Expert-verified Found in: Page 54 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Give a geometric interpretation of the linear transformations defined by the matrices in Exercises ${\mathbf{16}}$ through ${\mathbf{23}}$. Show the effect of these transformations on the letter ${\mathbit{L}}$ considered in Example ${\mathbf{5}}$. In each case, decide whether the transformation is invertible. Find the inverse if it exists, and interpret it geometrically. See Exercise ${\mathbf{13}}$$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$

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

The geometrical interpretation is: See the step by step solution

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

Let the matrix be,

$\mathrm{T}\left(\stackrel{\to }{\mathrm{x}}\right)=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\stackrel{\to }{\mathrm{x}}$

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

## Step 2: Compute the vectors.

Let the matrix be,

$\mathrm{T}\left(\stackrel{\to }{\mathrm{x}}\right)=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\stackrel{\to }{\mathrm{x}}$

Consider the vector $\left[\begin{array}{c}1\\ 0\end{array}\right]$

$\begin{array}{rcl}\mathrm{T}\left(\stackrel{\to }{\mathrm{x}}\right)& =& \left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\stackrel{\to }{\mathrm{x}}\\ & ⇒& T\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]\\ \therefore T\left[\begin{array}{c}1\\ 0\end{array}\right]& =& \left[\begin{array}{c}1\\ 0\end{array}\right]\end{array}$

Consider the vector $\left[\begin{array}{c}0\\ 2\end{array}\right]$

$\begin{array}{rcl}\mathrm{T}\left(\stackrel{\to }{\mathrm{x}}\right)& =& \left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\stackrel{\to }{\mathrm{x}}\\ & ⇒& T\left[\begin{array}{c}0\\ 2\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}0\\ 2\end{array}\right]\\ \therefore T\left[\begin{array}{c}0\\ 2\end{array}\right]& =& \left[\begin{array}{c}0\\ 0\end{array}\right]\end{array}$

## Step 3: Graph the letter using matrix.

Now, graph the original vectors and the obtained vectors as follow: $T\stackrel{\to }{\left(x\right)}$ is obtained by rotating the vector $\stackrel{\to }{x}$ 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 $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is invertible if and only if ${\mathbit{a}}{\mathbit{d}}{\mathbf{-}}{\mathbit{b}}{\mathbit{c}}{\mathbf{\ne }}{\mathbf{0}}$${\mathbit{a}}{\mathbit{d}}{\mathbf{-}}{\mathbit{b}}{\mathbit{c}}{\mathbf{\ne }}{\mathbf{0}}$.

The inverse of the matrix ${\mathbf{\left[}}\begin{array}{cc}\mathbf{a}& \mathbf{b}\\ \mathbf{c}& \mathbf{d}\end{array}{\mathbf{\right]}}$ is, ${{\mathbf{\left[}}\begin{array}{cc}\mathbf{a}& \mathbf{b}\\ \mathbf{c}& \mathbf{d}\end{array}{\mathbf{\right]}}}^{\mathbf{-}\mathbf{1}}{\mathbf{=}}\frac{1}{ad-bc}{\mathbf{\left[}}\begin{array}{cc}\mathbf{d}& \mathbf{-}\mathbf{b}\\ \mathbf{-}\mathbf{c}& \mathbf{a}\end{array}{\mathbf{\right]}}$ .

Consider the matrix,

$T=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒\left(1×0\right)-\left(0×0\right)=0$

Therefore, the given matrix is non-invertible and the shape of L gets transformed as a straight line along x-axis. ### Want to see more solutions like these? 