Definition of direct sum over MANY subspaces: Let $V$ be a vector space over a field $F$ and let $W_i~~~(1 \leq i \leq s;s \geq 2)$

Question

Definition by my professor: Let $V$ be a vector space over a field $F$ and let $$W_i~~~(1 \leq i \leq s;s \geq 2)$$ be vector subspaces of $V$. We say that the sum $W_1+W_2+...+W_s$ is a direct sum of vector subspaces $W_1,...,W_s$, if the intersection $$\left(\sum_{i=1}^{k-1}W_i\right) \cap W_k = \{0\},~~~(2 \leq \forall k \leq s)$$

I got confused with this "not so straightforward definition (mostly cause notation wise)". After trying to understand from most sources, i believe most of the definition likes to begin with the unique expression condition but my professor decided to go the opposite (it doesnt really matter right?)

So after taking a while to figure out, am i right to say that our professor meant that we need to check for all $k$ in between $2$ and the final subspace $W_s$ such that $$W_1\cap W_2 = \{0\}$$ $$(W_1+W_2)\cap W_3 = \{0\}$$ $$(W_1+W_2+W_3)\cap W_4 = \{0\}$$ $$\vdots$$ $$(W_1+W_2+...+W_{s-1}) \cap W_s = \{0\}$$

So essentially if we want to show that $W_1,W_2,W_3,W_4$ is a direct sum, we actually need to show that $W_1 \cap W_2 =\{0\}$, $(W_1+W_2) \cap W_3 = \{0\}$ and finally $(W_1+W_2+W_3) \cap W_4 = \{0\}$??? So many things to show?

Answer 1

• The "unique expression as a sum of elements of $W_1,\ldots, W_s$" definition is equivalent to the definition given by your professor, I believe.
• Your interpretation of the definition is correct.
• The intuition behind your professor's definition: You start with $W_1$ and sequentially "add" $W_2$, $W_3$, and so on, to build $W_1 + \cdots + W_s$. At each step, the empty intersection condition ensures that when you add $W_k$ to $W_1 + \cdots + W_{k-1}$, you are adding only "new" vectors (except the zero vector). If you think about it, this also ensures the "unique expression as a sum of elements of $W_i$" condition, since otherwise if, say, $W_k$ contained nonzero vectors of $W_1 + \cdots + W_{k-1}$, then there will be multiple ways to write such vectors as a sum of elements of $W_1,\ldots,W_s$.
• Your professor's definition may seem like a lot of conditions, but the "unique expression as a sum of elements of $W_1,\ldots, W_s$" definition is as hard to check (in general).