6.5. Exercises

State the matrices associated to the following linear transformations:

1. (i)

   $T\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)=\left(\begin{array}{c}\hfill x\hfill \\ \hfill 4y-x\hfill \end{array}\right)$

2. (ii)

   $T(x,y,z)=(5y+z,x-y)$

3. (iii)

   $T(a,b)=(2b-a,a+3b)$

4. (iv)

   $T(s,t)=(s,s+t,t-s)$

5. (v)

   $R_{\pi /4}$

6. (vi)

   $H_{\pi /4}$

Find the images of the points

by the following linear transformations $T$.

1. (i)

   $T\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)=\left(\begin{array}{c}\hfill x+2y\hfill \\ \hfill 3x+4y\hfill \end{array}\right).$

2. (ii)

   $T\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -3y\hfill \\ \hfill 2x-y\hfill \end{array}\right)$.

3. (iii)

   $T\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 3x+y\hfill \\ \hfill -x+y\hfill \end{array}\right)$.

4. (iv)

   $T$ is given by the matrix .

5. (v)

   $T$ is given by the matrix .

6. (vi)

   $T$ is the anticlockwise rotation through $\theta =\frac{2\pi }{3}$.

7. (vii)

   $T$ is the reflection about the $x$-axis.

8. (viii)

   $T$ is the reflection about the line which makes an angle $\frac{\pi }{6}$ above the positive $x$-axis.

9. (ix)

   $T$ is the composition $R_{\pi /4}{H}_{\pi /3}$.

10. (x)

    $T$ is the composition $H_{\pi /3}{R}_{\pi /4}$.

Prove that the identity matrix $I_n$ is the matrix of the identity map

If you compose two reflections, you get a rotation. If you compose a reflection with a rotation, you get a reflection. With this in mind, solve the following matrix equations for $\theta$. If you’ve already solved Exercise 6.5.5, then this question should be easy.

1. (i)

   $H_{\pi /4}{R}_{\pi /4}=H_{\theta }$.

2. (ii)

   $R_{\pi /4}{H}_{\pi /4}=H_{\theta }$.

3. (iii)

   $H_{\theta }{R}_{\pi /2}=H_{-\pi /4}$.

4. (iv)

   $H_{2\pi /3}{H}_{\pi /3}=R_{\theta }$.

5. (v)

   $H_{\theta }{H}_{\pi /4}=R_{\pi /6}$.

Let $a,b,c$ be angles. Find an equation relating angles $a,b,$ and $c$ (up to adding of multiples of $2\pi$) in the following cases:

1. (i)

   $H_a{H}_b=R_c$,

2. (ii)

   $H_a{R}_b=H_c$,

3. (iii)

   $R_a{H}_b=H_c$.

Recall that $H_a{H}_a=Id$ and $R_a{R}_{-a}=Id$.

Let $l,l^{\prime }$ be two lines in ${ℝ}^2$ which intersect in a point $P$, and $T$ a linear transformation of ${ℝ}^2$. Prove that $T(P)$ is the intersection of $T(l)$ and $T(l^{\prime })$.

Write the matrices of the following linear transformations of ${ℝ}^3$.

1. (i)

   A reflection in the $xz$-plane, which maps $e_1$ and $e_3$ to themselves, and maps $e_2$ to $-e_2$.

2. (ii)

   The rotation around the $x$-axis by the angle $\frac{\pi }{4}$ which sends the positive $y$-axis towards the positive $z$-axis.

3. (iii)

   For $\lambda \in ℝ$, consider the linear transformation $Tv=\lambda v$. For which values of $\lambda$ is this non-invertible?

For each of the following matrices $A$ consider its associated linear transformation $T$.

• (a)

  Find all $a\in ℝ$ such that $T$ is an invertible linear transformation of ${ℝ}^n$, and

• (b)

  Find $T^{-1}$ in terms of $a$, for the values of $a$ obtained in part (a).

1. (i)

2. (ii)

3. (iii)

4. (iv)

5. (v)

In each of the following questions, find the image of the line $l$ by the invertible linear transformation $T$ given by the matrix $A$.

1. (i)

   $l:y=\frac{2}{3}x-1$ and .

2. (ii)

   $l:x+2y-6=0$ and $A=R_{\pi /6}$.

3. (iii)

   $l:y=x+8$ and .

Let $l_{\theta }$ be the line in ${ℝ}^2$ going through $(0,0)$ which makes an angle $\theta$ above the positive $x$-axis. Consider the linear transformation associated to the matrix:

This is called the orthogonal projection onto the line $l$.

1. (i)

   For which values of $\theta$ is this an invertible linear transformation?

2. (ii)

   For $\theta =0$, write the matrix of ${\pi }_0$. How does ${\pi }_0$ transform ${ℝ}^2$? You may draw a picture, or explain in words.

3. (iii)

   By multiplying the matrices, show that ${\pi }_{\theta }=R_{\theta }{\pi }_0{R}_{-\theta }$.

4. (iv)

   For any point $v\in {ℝ}^2$, show that ${\pi }_{\theta }(v)$ lies on the line $l_{\theta }$.

5. (v)

   The set of all points in ${ℝ}^2$ which get mapped to $0$ by ${\pi }_{\theta }$ is a line. Prove that this line is perpendicular to $l_{\theta }$.

Consider the unit square in ${ℝ}^2$ whose vertices are at $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. The matrix transforms this square into a parallelogram.

1. (i)

   Find the area of that parallelogram.

2. (ii)

   Any matrix $A\in M_3(ℝ)$ transforms the unit cube in ${ℝ}^3$ to a new shape. Can you guess what the volume of that shape is? [Hint: First try part (i)]

Consider the set of six linear transformations:

Prove that if you compose any two transformations from $G$, the resulting transformation is also in $G$. For example, $R_{2\pi /3}{R}_{4\pi /3}=R_0$.

A structure like $G$, is said to be closed under multiplication, and is called a group (as studied in Group theory). Since these are linear transformations of ${ℝ}^2$, this is also an example of a 2-dimensional representation of the abstract group $G$ (as studied in Representation theory).