Hi Lul i will speak english in order to don't make a mistake.
Is my procedure correct?
first of all i proved that f (in this case f2 ) is linear by proving f (λx + μy) = λf(x) + μf(y)
Now i want to calculate Ker (f) by saying that f(x) = the0vector
this leads to a linear system
{x1 + x2 = 0 and
{ x1 + x2 + x3 + x4 = 0
resolving the system leads me to x1 = -x2 and x3 = -x4
1. Question is this the value of ker(f) ? if not how i calculate tha value of ker(f)? on Youtube i can only find videos showing what's the procedure to calculate the dim(ker(f)).
About the dimension:
i then wrote the linear system as the associated matrix in this case:
\( \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{pmatrix} \)
i then used the gauss algoritmus to have 0s at the bottom of the matrix and i found that the reduced matrix:
\( \begin{pmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix} \)
has 1 pivot ⇒ so the dim(ker(f)) should be 1 right?
if not can you please explain what i did wrong?
2/3. Question: In the excersise is not requested to find a basis for the kernel but you wrote a procedure to calculate it. Should i do it any time that is requested to find the dimension of the kernel?