Linear application $\mathbb{R}^4 \to \mathbb{R}^3 $

Question

I have this exercise that I cannot solve even though really thinking about it. Consider $\mathbb{R}^4$ and $\mathbb{R}^3$ as vector spaces. Let $f: \mathbb{R}^4 \to \mathbb{R}^3 $ be the linear application defined by $f(x_1,...,x_n)^T=A (x_1,...,x_n)$ where $ A=\begin{pmatrix} 1 & 2 & 1 & 0 \\ 1 & 0 & 1 & 1 \\ -1 & 0 & 0 & 1 \\ \end{pmatrix} $

i) Prove that $f$ is onto

ii) Prove that if $v_1$ and $v_2 $ are two preimages of $u$ then $v_1 - v_2 \in Ker(f) $

iii) Prove that if $v$ is a preimage of $u$ and $v_0 \in Ker(f)$ then $v+v_0$ is also a preimage of $u$

I will be very thankful to your answers.

Answer 1

Hints:

1) In this case, $\;f\;$ is onto iff $\;\text{rank}\,A=3\;$ , and remember: the row rank and the column rank of a matrix are equal...

2) Use linearity of $\;A\;$ : $\;Av_1-Av_2=A(v_1-v_2)\;$

3) Exactly as (2) above: linearity.