Let $V$ be the subspace of $\mathbb{R}^5$, spanned by the vectors a, b and c, with
$$\textbf{a} = (2,1,3,1,-1),\hspace{1mm}b=(4,-3,5,1,1),\hspace{1mm}c=(-1,2,-2,7,3)$$
(a) Use any test you like to determine if the given vectors are linearly independent.
(b) From among the vectors a, b and c, select an ordered basis $\mathfrak{B}$ for $V$. (You need to prove that vectors you propose actuually form a basis for $V$).
I row reduced the matrix whoes columns are a,b,c and I only have the trivial solution for the system $$c_1\textbf{a}+c_2\textbf{b}+c_3\textbf{c}=0$$
which means a,b,c are linearly independent. I'm confused about part (b) because from what I have a bases for $V$ would be $\mathfrak{B}=\lbrace\textbf{a},\textbf{b},\textbf{c}\rbrace$.
am I missing something because part (b) of the question is worth 10 points and it seems I don't have to do anything to solve it.
From row reducing the matrix with ROWS $a$,$b$ and $c$ you get that the rank of the matrix is $3$ which means that $a$,$b$ and $c$ are linearly independent. For the second question,as you mentioned,all you have to say is that $B = {(a,b,c)} $ because you know that $a$,$b$ and $c$ are independent and span the vector $V$.