I want to prove that:
If $W_1$ and $W_2$ are finite-dimensional subspaces of a vector space $V$, then the subspace $W_1 + W_2$ is finite-dimensional.
I have written a skeleton of a proof but I don't know how to complete it. Could you guys look it over and tell me what's wrong about it? Thanks:
We know the intersection of two subspaces is also a subspace. So, $W_1 \cap W_2$ is a subspace. Since $W_1$ and $W_2$ are both finite-dimensional, their intersection must also be finite-dimensional. Thus, $W_1 \cap W_2$ is finite-dimensional. Because subspaces are, themselves, vector spaces we know that a finite-dimensional vector space includes a basis consisting of a finite number of vectors. So, $W_1 \cap W_2$ has a basis $\beta = \{u_1, \cdots, u_n\}$ with $n < \infty$. We know $W_1 \cap W_2 \subseteq W_1$ So, we can extend $\beta$ to a basis for $W_1$ which we will call $\gamma = \{u_1, \cdots , u_n, p_1, \cdots , p_m \}$. Similarly, since $W_1 \cap W_2 \subseteq W_2$ we can extend $\beta$ to a basis of $W_2$ which we will call $\alpha = \{u_1, \cdots , u_n, v_1, \cdots , v_k \}$. By definition of a basis, we know $\gamma$ spans $W_1$ and $\alpha$ spans $W_2$. Therefore, $\text{span}(W_1\cup W_2) = \text{span}(\gamma) + \text{span}(\alpha)$. But, $W_1 + W_2 = \text{span}(W_1 \cup W_2)$ by definition implying $W_1 + W_2 = \text{span}(\gamma) + \text{span}(\alpha)$. Since $\text{span}(\gamma)$ and $\text{span}(\alpha)$ are both finite-dimensional, $W_1 + W_2$ must also be finite-dimensional.
That is my proof, but I am still a little worried about the last few parts. Is it true that the span of the union of two subspaces is exactly the sum of the spans of the two subspaces? Thank you!
Best Answer
Since this is such basic stuff, you must really be clear about what definitions you are using. In the courses I know (and give) "finite dimensional" is actually defined before dimension is, and it means "finitely generated", in other words a space is finite dimensional if (and only if) it is the span of a finite set. Span of a set of vectors is formed by the set of their linear combinations, which forms the smallest subspace containing that set. From this it is easy to show that the span of $A\cup B$ is the sum of the span of $A$ and the span of $B$ (which sum can be defined as the span of their union). Once this is done, it is clear that the sum of two finite dimensional subspaces (spans of finite $A,B$ respectively) is finite dimensional (the span of the finite set $A\cup B$).
On the other hand it is not so obvious (though true) that any subspace of a finite dimensional space is finite dimensional, and in particular that the intersection of a finite dimensional subspace with another subspace (which may or may not be finite dimensional) is finite dimensional.