Intuition behind the definition of direct sum of linear subspaces I'm reviewing linear algebra reading the excellent book Linear Algebra by Hoffman and Kunze and they define the direct sum of subspaces as one of these equivalent statements on page 209:

Lemma. Let V be a finite-dimensional vector space. Let $W_1,\ldots , W_k$ be subspaces of V and let $W= W_1+\ldots+W_k$. The following
are equivalent.
(a) $W_1,\ldots,W_k$ are independent.
(b) For each $j$, $2\le j \le k$, we have $W_j \cap (W_1 + . . . +
 W_{j-1}) = \{0\}$
(c) If $\mathscr B_i$ is an ordered basis for $W_i$, $1\le i \le k$,
then the sequence $\mathscr B = (\mathscr B_1,\ldots , \mathscr B_k)$
is an ordered basis for $W$.

I want to know if there is a geometric intuition behind this definition. I've already seen some intuitions in some specific cases with some very specific subspaces. I've been thinking if there is a more general intuition behind the concept of direct sums.
 A: So I'm far from an expert, hence this is just a try. But maybe we can think for example of a two dimensional subspace in euclidean space, thus a plane, and a one dimensional subspace that is not contained in the plane. This says that the only intersection of the plane and the line is the origin, hence their sum is "direct". Then by the dimension formula their sum is the whole space, but we can also see this geometrically when we "move" the plane along the line. By this, we will reach every point of the space. And that this sum is direct may mean that we may reach every point uniquely - I think of this right now as moving the plane along the line, the line may be viewed as a scale, and at no two points of this scale the planes at this position intersect.
I just think of a direct sum as patching together subvector spaces that give the total space, but in a some sense in a minimal way. This is what your definition (c) precisely is about: that we get an ordered generating system means that the subvector spaces "patched together" give the whole space, and that it is a basis - so also linearly independent - means that it's minimal. Maybe we can think of diviging the whole space in smaller pieces that have information about the whole space; that the sum of the vector-space is the whole space means that the whole information in all the sub-vector-spaces gives all the information, that the sum is direct means that no information occurs "doubled" in different sub-vector-spaces
I hope this was in some way helpful.
