Definition and examples

Consider the following functions $T:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}$. For each of these functions $T$, show that it is not linear either by showing that $T$ does not preserve addition, or by showing that it does not preserve scalar multiplication, or by showing that it does not preserve the zero vector.

1. $T\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}x+2y+3z+1\\ 2y-3x+z\end{array}\right]$.

2. $T\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}x+2y^{2}+3z\\ 2y+3x+z\end{array}\right]$.

3. $T\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}\sin x+2y+3z\\ 2y+3x+z\end{array}\right]$.

Which of the following functions are linear transformations?

1. The function $f:\mathbf{Seq}_{K}\to\mathbf{Seq}_{K}$ defined by $f((a_{0},a_{1},a_{2},\ldots))=(-a_{1},-a_{3},-a_{5},\ldots)$. In words, the function $f$ removes the even-numbered elements, and negates the odd-numbered elements.

2. The function $f:\mathbf{M}_{2,2}\to\mathbf{M}_{2,2}$ defined by $f(A)=AB$. Here, $\mathbf{M}_{2,2}$ is the vector space of $2\times 2$-matrices with real entries, and $B$ is a fixed matrix.

3. The function $f:\mathbf{M}_{2,2}\to\mathbf{M}_{2,2}$ defined by $f(A)=A+B$, where $B$ is a fixed matrix.

4. The function $f:\mathbf{M}_{2,2}\to\mathbf{M}_{2,2}$ defined by $f(A)=A^{T}$.

Consider the shift function $\operatorname{shift}:\mathbf{Seq}_{K}\to\mathbf{Seq}_{K}$ from Example 10.8. Find a basis for the solution space of each of the following equations:

1. $\operatorname{shift}(a)=a$.

2. $\operatorname{shift}(a)=-a$.

3. $\operatorname{shift}(\operatorname{shift}(a))=\operatorname{shift}(a)+2a$.