I am newbie started learning the linear algebra. It might be dumb question. But I don't understand how the sum of subspace can also be subspace?!
So for subset in order to be subspace, It should satisfy 3 features:
- Includes $0$
- Closed under $+$
- Closed under $\times$
I understand it.
But when it comes to sum of different two subspaces, the feature 2 above doesn't seem to make sense to me.
Suppose $U,W=\Bbb{R}^2$ (2 dimentions). $$U=\{(x, y) \in \Bbb{R}^2 : y=0\}, \quad W=\{(x, y) \in \Bbb{R}^2 : x=0\}.$$ Then $U+W=S$ is, in my intuitive thought going to be a cross only including all vectors along the origin lines($x$-axis and $y$-axis from $0$). Then let's check if $S$ satisfies the addition feature. $$(2,0) \in S, \quad (0,2) \in S,$$ then the addition vector of these two vector is not in $S$.
So it is quite confusing to me. Maybe I'm taking a simple concept in the wrong way. Hope somebody can help me.
You are right to notice that closure under addition might be missing if you only take the union of two subspaces. In general, the union of two subspaces is not a subspace. We can fix this by "filling in the gaps," which amounts to introducing all linear combinations of elements in both subspaces.
A good definition for this is to take the union of $U$ and $W$, and then take the span of the resulting union. Convince yourself that this fixes the problem!