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I have to find out the dimension of the solution of this set of the equations with the system being an affine space of $\Bbb R^4$ and $t∈R$, $u∈\Bbb R^4$.

So I tried to row reduce the left side of the system and found out that the rank is $4$ if $t≠0$ and $3$ if $t=0$. With the formula $\dim(L)=n-\operatorname*{rank}(A)$, I think that the dimension is $1$ for $t=0$ or $0$ for $t≠0$. But I am completely unsure if any of this is correct.

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How does your set of solution looks like?

I think you are right, but I am unsure because of your statement for t = 0, because you can reduce further than just row echelon form until you reach the 4 canonical basis vectors of $\mathbb{R}^4$

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    http://www.wolframalpha.com/input/?i=row+reduce+%7B%7Bt*a,(t%2B1)*b,0,0%7D,%7B0,t*b,(t-1)*c,0%7D,%7B0,(t-1)*b,t*c,0%7D,%7Ba,0,(t%2B1)*c,t*d%7D%7D thats the wolframalpha link for the reduced form it also includes t=1/2, so my first step would be to calculate the determinant of A and then try to reduce the the set of equations for my different values of t? – 2017-01-26