Exercise 10.1.3

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 $f(A)=A^{T}$.