2 subspaces of R^4, bases, intersection

Hi MHF,

I appreciate very much all the help I got so far. If I didn't answered my last 2 posts it's because of lack of time. I feel overloaded with linear Algebra and the final exam is coming up next 26th of February.

Here is the exercise I'd like to check out with you.

Let $\displaystyle S_1=\{ (1,0,1,2), (-1,2,3,8), (-1,1,1,3), (0,1,2,5) \}$ and $\displaystyle S_2=\{ (1,1,0,1), (1,-1,0,3), (3,1,2,4), (1,1,2,0) \}$ and let $\displaystyle W_1$ be spanned by $\displaystyle S_1$ and $\displaystyle W_2$ be spanned by $\displaystyle S_2$.

a)Give a base $\displaystyle B_i$ of $\displaystyle W_i$ contained in $\displaystyle S_i$, for i=1, 2.

My attempt : I wrote each vector of the base into a matrix as being column vectors and reduced it. I got that 3 of the 4 vectors of $\displaystyle S_1$ were colinear and then I solve a 2 equations system, getting finally that a base of $\displaystyle W_1$ included into $\displaystyle S_1$ is $\displaystyle \{ (1,0,1,-1), (0,1,-1,1) \}$.

Doing the same for $\displaystyle S_2$, I got that a base of $\displaystyle W_2$ included into $\displaystyle S_2$ is $\displaystyle \{ (1,0,0,0), (0,-1,1,-1) \}$.

b)Describe $\displaystyle W_1$, $\displaystyle W_2$, $\displaystyle W_1 \cap W_2$ and $\displaystyle W_1+W_2$ implicitly.

My attempt : don't know how to start.

c)Give a base of $\displaystyle W_1 \cap W_2$ and of $\displaystyle W_1+W_2$.

My attempt : I wrote the bases I got in part a) as vector column in a matrix. I checked out that $\displaystyle \det A \neq 0$, hence the matrix is invertible and the 4 vectors are linear independent. So I think that $\displaystyle W_1 \cap W_2 = $∅. So there's no base that generates $\displaystyle W_1 \cap W_2$, or I'm wrong?

And as the 4 vectors are linear independent, a basis of $\displaystyle W_1+W_2$ would be the union of the bases of $\displaystyle W_1$ and $\displaystyle W_2$. Or : $\displaystyle \{ (1,0,1,-1), (0,1,-1,1),(1,0,0,0), (0,-1,1,-1) \}$. Am I right?