Exercise 21

Prove that for any linear map $h\colon V\to W$ and any $\vec{w}\in W$, the set $h^{-1}(\vec{w})$ has the form

(if $h$ is not onto and $\vec{w}$ is not in the range of $h$ then this set is empty since its third condition cannot be satisfied). Such a set is a coset of $\mathscr{N}(h)$ and we denote it as $\vec{v}+\mathscr{N}(h)$.

Consider the map $t\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ given by

for some scalars $a$, $b$, $c$, and $d$. Prove that $t$ is linear.

Conclude from the prior two items that for any linear system of the form

we can write the solution set (the vectors are members of $\mathbb{R}^{2}$)

where $\vec{p}$ is a particular solution of that linear system (if there is no particular solution then the above set is empty).

Show that this map $h\colon\mathbb{R}^{n}\to\mathbb{R}^{m}$ is linear

for any scalars $a_{1,1}$, …, $a_{m,n}$. Extend the conclusion made in the prior item.

Show that the $k$-th derivative map is a linear transformation of $\mathcal{P}_{n}$ for each $k$. Prove that this map is a linear transformation of the space

for any scalars $c_{k}$, …, $c_{0}$. Draw a conclusion as above.