1.
\(T\left[\left(\begin{array}{c}x\\y\end{array}\right)\right]=\left(\begin{array}{c}3x-y\\4x-y\end{array} \right)\)
Section 9.3 Linear Transformations: Homework
Exercises Exercises
Exercise Group.
For each of the following,
- Determine with proof whether the mapping \(T\) is a linear transformation.
-
If \(T\) is a linear transformation,
- Find a matrix \(A\) so that \(T(\vec{x})=A\vec{x}\text{.}\)
- Find the range of \(T\text{.}\)
- Find the kernel of \(T\text{.}\)
- Determine whether \(T\) is injective.
- Determine whether \(T\) is surjective.
2.
\(T\left[\left(\begin{array}{c}x\\y\end{array}\right)\right]=\left(\begin{array}{c}x+3y\\-2x+6y\end{array} \right)\)
3.
\(T\left[\left(\begin{array}{c}x\\y\\z\end{array}\right)\right]=\left(\begin{array}{c}3x\\4y\\x-y-1\end{array} \right)\)
4.
\(T\left[\left(\begin{array}{c}x\\y\\z\end{array} \right)\right]=\left(\begin{array}{c}-x+3z\\y\\x-2y-z \end{array} \right)\)
5.
\(T\left[\left(\begin{array}{c}x\\y \end{array} \right)\right]=\left(\begin{array}{c}2y\\x^2 \end{array} \right)\)
6.
\(T\left[\left(\begin{array}{c}x\\y \end{array} \right)\right]=\left(\begin{array}{c}x+y\\0\\ y \end{array} \right)\)
7.
Find a matrix that represents the linear transformation of vectors in \(\R^2\) which rotates each vector counterclockwise about the origin by \(\pi/4\) radians.
8.
Find a matrix that represents the linear transformation of vectors in \(\R^2\) which reflects each vector about the \(x-\)axis.
9.
The population of male and female wolves of an endangered species in a certain region varies each year according to a linear transformation \(T\text{.}\) Denote by \(m\) the number of males and \(f\) the number of females, and let \(\vec{v}=\left(\begin{array}{c}m\\ f\end{array} \right)\text{.}\) After one year the population is \(T(\vec{v})\text{.}\) Suppose \(T\left[\left(\begin{array}{c}3430\\1590 \end{array} \right)\right]=\left(\begin{array}{c}1510\\700 \end{array} \right)\) and \(T\left[\left(\begin{array}{c}1510\\700 \end{array} \right)\right]=\left(\begin{array}{c}760\\250 \end{array} \right)\text{.}\)
- Find a matrix \(A\) so that \(T(\vec{v})=A\vec{v}\text{.}\)
- Will the population ever be \(m=600, f=200\text{?}\) Explain how you know.
10.
A game is played where \(3\) values are input to a black box which then returns \(3\) outputs. If you put in \(3\) inputs and get the same \(3\) outputs you win \(\$10,000\text{.}\) We are trying to break the code and figure out how the black box operates. We observe the following three input / output pairs:
\begin{equation*}
\left(\begin{array}{r} 1\\1\\2 \end{array} \right) \rightarrow \left(\begin{array}{r} 3\\0\\0 \end{array} \right), \left(\begin{array}{r} 0\\-1\\0 \end{array} \right) \rightarrow \left(\begin{array}{r} 0\\1\\0 \end{array} \right), \left(\begin{array}{r} 1\\0\\1 \end{array} \right) \rightarrow \left(\begin{array}{r} 0\\0\\2 \end{array} \right)
\end{equation*}
If we assume the black box performs some linear transformation on the inputs, then
- Determine a matrix \(A\) that represents this linear transformation.
- Find a set of “winning values” (for which the input will equal the output) or explain why this is not possible.
