Given a linear function, how can I find the Ker?

Question

Given the linear function

$f : (a, b, c, d, e) ∈ R^5 → (a − d, a − d, e − d, c) ∈ R^4$

How can I find the Ker?

Please help me.

Answer 1

Equal all the coordinates to zero...:

$$\begin{cases}a-d=0\\ a-d=0\\e-d=0\\c=0\end{cases}\;\;\implies a=d=e\;,\;\;c=0$$

so

$$\ker f=\left\{ (a,b,0,a,a)\;|\;a\in\Bbb R\right\}$$

Answer 2

Given $f:R^n\to R^m $ note that $\ker f$ is defined as the set of all $x\in R^n$ such that $f(x)=0$, in this case we have

$$f(a, b, c, d, e)=(a − d, a − d, e − d, c)=0\implies\begin{cases}a-d=0\\a-d=0\\e-d=0\\c=0\end{cases}\implies\begin{cases}a=d\\e=d\\c=0\end{cases}$$

thus set $d=t\in R$ and $b=s\in R$ we have

$$\ker f=\{(t,s,0,t,t)\in R^5 \quad t,s\in R\}$$

and $\ker f$ has dimension 2 since you have 2 free parameters and a basis is

$$v_1=(1,0,0,1,1) \quad v_2(0,1,0,0,0)$$