Linear Algebra with Applications, 10th Edition

Section 4.1 Exercises

1. Show that each of the following are linear operators on ${\mathrm{ℝ}}^2$. Describe geometrically what each linear transformation accomplishes.

   1. $L\left(\mathbf{x}\right)={\left(-x_1,x_2\right)}^T$

   2. $L\left(\mathbf{x}\right)=-\mathbf{x}$

   3. $L\left(\mathbf{x}\right)={\left(x_2,x_1\right)}^T$

   4. $L\left(\mathbf{x}\right)=\frac{1}{2}\mathbf{x}$

   5. $L\left(\mathbf{x}\right)=x_2{\mathbf{\text{e}}}_2$

2. Let L be the linear operator on ${ℝ}^2$ defined by

   $$L\left(\text{x}\right)={\left(x_1\cos \alpha -x_2\sin \alpha , x_1\sin \alpha +x_2\cos \alpha \right)}^T$$

   Express $x_1,x_2$, and $L\left(\text{x}\right)$ in terms of polar coordinates. Describe geometrically the effect of the linear transformation.

3. Let a be a fixed nonzero vector in ${\mathrm{ℝ}}^2$. A mapping of the form

   $$L\left(\text{x}\right)=\mathbf{x}+\mathbf{a}$$

   is called a translation. Show that a translation is not a linear operator. Illustrate geometrically the effect of a translation.

4. Let $L:{\mathrm{ℝ}}^2\to {\mathrm{ℝ}}^2$ be a linear operator. If

   $$L\left({\left(1,2\right)}^T\right)={\left(-2,3\right)}^T$$

   and

   $$L\left({(1,-1)}^T\right)={(5,2)}^T$$

   find the value of $L({(\mathrm{7,\; 5})}^T)$.