How to determine basis and dimension of subspaces?

Question

I'm new into linear algebra and I encountered the following problem:

In $\mathbb{R}^4$ we conside the following subspaces: $U=\{(x.y.z.w)\in \mathbb{R}^4 | x+y+z=0\}$, $V=\mathscr{L}((1,1,0,0), (2,-1,0,1),(4,1,0,1)).$

1. Determine the basis and dimensions of U, V and an algebraic representation of V.

2. Determine the basis and dimensions of $U \cap V$ and $U+V$.

3. Complete the base of $U \cap V$ in a base of $\mathbb{R}^4$.

Firstly, I know that the subspace U is the set of solutions of the linear equation $x+y+z=0$ . But now I'm stuck as I have never seen such type of problem and don't really understand what should I do.

Can someone explain in detail the steps?

Thanks.

Answer 1

HINT

Firstly note that $U$ has dimension $3$ since we have three free variables, thus to find a base we can consider the three linearly independent vectors

• $(x,y,z,w)=(1,-1,0,1) $
• $(x,y,z,w)=(1,0,-1,1)$
• $(x,y,z,w)=(0,1,-1,1)$

Can you proceed from here?