I'm trying, in vain, to find the basis of the following vector spaces:
(a) $W = \{x = (x_1 , x_2 , x_3 ) ∈ \Bbb R^3 : x_1 − 2x_2 + x_3 = 0, 2x_1 − 3x_2 + x_3 = 0\}$
(b) $W = \{x = (x_1 , x_2 , x_3 , x_4 , x_5 ) ∈ \Bbb R^5 : x_1 − x_3 − x_4 = 0\}$
(c) $W = \{x = (x_1 , x_2 , x_3 , x_4 , x_5 ) ∈ \Bbb R^5 : x_2 = x_3 = x_4 , x_1 + x_5 = 0\}$
I understand that if I have a vector space $V$, then the basis $\mathcal B$ for that vector is the set of vectors which is linearly independent and spans all $V$.
However, how do I apply this to solve the questions above?
Thanks.
Edit: I tried to solve the second one which seems easier, but the problem I have is I could get a basis of 5 linearly independent vectors for it, but it's not the right answer. I'm not sure what I'm doing wrong. Here is what I got for (b):
$\mathcal B(V) = \{(1,0,0,0,0), (0,1,0,0,0), (0,0,1,0,0), (0,0,0,1,0), (0,0,0,0,1)\}$