Challenge Problems for Chapter 6

Argue geometrically to prove that the following transformations are linear:
1.
Rotation of the plane about the origin through angle \(\theta \).
2.
Reflection of the plane about the line \(y=mx\).
Think in terms of linearity diagrams.

[Picture]

[Picture]

What happens when you rotate two vectors first, then add them, versus adding the two vectors first, then rotating the sum?

The figure below illustrates the left side of the diagram. Vectors \(\vec {u}\) and \(\vec {v}\) are added in the domain, then the sum is rotated through angle \(\theta \).

[Picture]

The next figure illustrates what happens when vectors \(\vec {u}\) and \(\vec {v}\) are rotated through angle \(\theta \), then their images are added together.

[Picture]

Because the diagonal of a parallelogram rotates with the parallelogram, it is clear that

\[R_{\theta }(\vec {u}+\vec {v})=R_{\theta }(\vec {u})+R_{\theta }(\vec {v})\]
Find the matrix of the linear transformation which rotates every vector in \(\mathbb {R}^{3}\) counter clockwise about the \(z\) axis when viewed from the positive \(z\) axis through an angle of 30\(^{\circ }\) and then reflects through the \(xy\) plane.

Click on the arrow to see answer.

\[ \left [ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array} \right ] \left [ \begin{array}{ccc} \cos \left ( \frac {\pi }{6}\right ) & -\sin \left ( \frac {\pi }{6}\right ) & 0 \\ \sin \left ( \frac {\pi }{6}\right ) & \cos \left ( \frac {\pi }{6}\right ) & 0 \\ 0 & 0 & 1 \end{array} \right ] = \left [ \begin{array}{ccc} \frac {1}{2}\sqrt {3} & -\frac {1}{2} & 0 \\ \frac {1}{2} & \frac {1}{2}\sqrt {3} & 0 \\ 0 & 0 & -1 \end{array} \right ] \]
Let \(T:\RR ^n\rightarrow \RR ^m\) be a linear transformation such that \(\text {ker}(T)=\RR ^n\). Find \(\text {im}(T)\).
Let \(\vec {v}\) be a non-zero vector in \(\RR ^n\). Given any vector \(\vec {w}\) in \(\RR ^m\), show there exists a linear transformation \(T:\RR ^n\rightarrow \RR ^m\) with \(T(\vec {v})=\vec {w}\).
Given \(\vec {y}\) in \(\RR ^n\), define \(S_{\vec {y}}:\RR ^n\rightarrow \RR \) by \(S_{\vec {y}}(\vec {x})=\vec {x}\dotp \vec {y}\) for all \(\vec {x}\) in \(\RR ^n\).
1.
Show that \(S_{\vec {y}}\) is a linear transformation.
2.
Show that every linear transformation \(T:\RR ^n\rightarrow \RR \) arises in this way (i.e. \(T=S_{\vec {y}}\) for some \(\vec {y}\) in \(\RR ^n\).)
Write \(S_{\vec {y}}(\vec {e}_i)=y_i\) for \(i=1,\dots , n\).

Bibliography

Some of these problems come from Chapter 5 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)

Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, pp. 272–315.

Some of these problems come from the end of Chapter 7 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)

W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, pp. 376–386.