Unsure of attempt to determine $\dim(W_1+W_2)$ and $\dim(W_1 \cap W_2)$ Let $V=\mathbb{R}^4$. $W_1$ is a subspace of $V$ spanned by vectors $a_1=(1, 2, 0, 1)$ and $a_2=(1,1,1,0)$. $W_2$ is a subspace of $V$ spanned by vectors $b_1=(1,0,1,0)$ and $b_2=(1,3,0,1)$. Determine $\dim(W_1+W_2)$ and $\dim(W_1 \cap W_2)$.
Attempt
The vectors $a_1$ and $a_2$ are linearly independent and span $W_1$, so they form a basis for $W_1$. Hence $\dim(W_1)=2$. The vectors $b_1$ and $b_2$ are linearly independent and span $W_2$, so they form a basis for $W_2$. Hence $\dim(W_2)=2$.
This is the step that I'm unsure of. I added vectors from $W_1$ and $W_2$ together, i.e. $a_1 + b_1, a_1 + b_2, a_2 + b_1, a_2 + b_2$ to form $$W_1+W_2=\left\{(2,2,1,1),(2,5,0,2),(2,1,2,0),(2,4,1,1) \right\}.$$
The first three vectors are linearly independent, but $$(2,4,1,1)= (2,5,0,2)+(2,1,2,0)-(2,2,1,1).$$ Hence there are only $3$ vectors that are linearly independent and span $W_1+W_2$, so $\dim(W_1+W_2)=3$.
We know that $\dim(W_1+W_2) = \dim(W_1) + \dim (W_2) - \dim(W_1 \cap W_2)$. Therefore $\dim(W_1 \cap W_2)=2+2-3=1$.
Question
Is my attempt correct?
Thank you for your time.
 A: Your attempt is correct, but note that your understanding of $W_1 + W_2$ might be wrong. By definition, $W_1 + W_2 = \{\vec{w}_1 + \vec{w}_2 | \vec{w}_i \in W_i, i=1, 2\}.$ However, This doesn't mean that the bases have to be added together. In other words, you can simply take $\{a_1,a_2,b_1,b_2\}$ as a spanning set for the space $W_1 + W_2$ and then toss out any linearly dependent vectors. The easiest way to do this is to row reduce the matrix whose columns are the corresponding basis vectors. For your example, the matrix will row reduce to (why? -- try it!)
$$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0& 0 & 1 & 0 \\ 0 & 0 & 0 & 0\end{pmatrix}.$$
This tells you that you have three linearly independent vectors and so the dimension of the $W_1 + W_2$ is 3.
Edit: another related question  that is useful to ask yourself: if $a_1, a_2, a_3, a_4$ are linearly independent, are $a_1 + a_2, a_1 - a_2, a_3 + a_4, a_3 - a_4$? How would you check? (hint: checking independence can always be reduced to row reducing a matrix)
