Dimension of Sum of Subspaces Proof The following is a proof I gave that was marked as fundamentally incorrect. If anyone could let me know where I went wrong that would be greatly appreciated. Note that I'm just outlining the main idea here. I was a little bit more careful in the actual argument. I'm just wondering if there is a major methodological flaw anywhere.
We want to prove that
$\dim(V  +W) = \dim(V) + \dim(W) - \dim(V \cap W)$
Proof.
Let $(v_1,...,v_n)$ be a basis for $V$. Then by definition $\dim V = n$. We can extend
this basis of $V$ to a basis of $V + W$ (proof given in class textbook). Thus let $(v_1,...,v_n,w_1,...,w_m)$ be a basis for $V + W$. Thus $\dim(V + W) = n + m$.
We know that we can reduce a basis of $V + W$ to a basis for $W$ (proof also given in textbook) by removing some number of basis vectors not in the span of $W$. We claim that $(v_1,...,v_r,w_1,...,w_m)$ is a basis for $W$ with $r \leq n$. We cannot remove vectors not in the span of a basis for $V$ or else they would not be in $V + W$. Thus we have $\dim W = r +m$. Since we removed $r$ vectors, $\dim(V \cap W) = r$. So
$\dim(V  +W) = n + (r +m) - r$
which means $\dim(V+W) = n +m$ and we're done.
Thanks very much in advance.
 A: Let's follow your proof with an example. Say we consider $\Bbb{R}^3$, and $V = \operatorname{span}((1, 0, 0), (0, 1, 0))$, i.e. the $xy$ plane, and $W = \operatorname{span}((0, 1, 0), (0, 0, 1))$, i.e. the $yz$ plane. Note that $V + W = \Bbb{R}^3$.
The first step is to pick a basis for $V$. Just to be difficult, I'll choose $((1, 1, 0), (1, -1, 0))$.
Next, we extend this basis of $V$ to a basis of $V + W = \Bbb{R}^3$, using vectors from $W$. I'll add the single vector $(0, 1, 1) \in W$. This gives us a basis
$$((1, 1, 0), (1, -1, 0), (0, 1, 1))$$
of $\Bbb{R}^3 = V + W$. So far, so good.
Now, your proof instructs me to reduce the above basis to a basis of $W$. Here's where we have a problem! We can't find enough vectors in the above basis that belong to $W$. Only that final vector $(0, 1, 1)$ belongs to $W$. The others don't, because they have a non-zero $x$ coordinate. So, the largest subset of $W$ that we can form from this basis has size $1$, which does match $W$'s dimension of $2$. We can't even make it span $W$!
Instead, you should instruct the reader to form a basis by considering a basis of $V \cap W$ first. You can then extend this to a basis of $V$, or to a basis of $W$. Think about how these bases might combine to give a basis for $V + W$.
