Could someone provide examples of sum and direct sum of vector subspaces?
Maybe with geometric vectors, row vectors, polynomials, anything…
Difference between sum and direct sum
The way I get it is that every direct sum is also a sum.
What is an example (if not too trivial even better) of a sum which is not a direct sum?
I have a book with a few definitions and theorems but not a single example related to these two concepts.
Note that by sum I don't mean union in set theory sense but sum of two subspaces as defined here:
Best Answer
The point of a direct sum is that each of the summands is independent of the others in the sense that if $V=\bigoplus_{k=1}^n U_k$, then every vector in $V$ can be uniquely expressed as a sum of exactly one vector per direct summand (in contrast to a non-direct sum, where the expression exists but is not unique). For instance, if we have two subspaces of $\mathbb R^3$ where $U_1$ is spanned by $(1,0,0)$ and $U_2$ is spanned by $(0,1,0)$ and $(0,0,1)$, then there is only one way to express $(a,b,c)$ as a sum of a vector from $U_1$ and a vector from $U_2$: it must be $(a,0,0)+(0,b,c)$. So $U_1+U_2$ is direct, or in short: $V=U_1\oplus U_2$.
A very simple example of a sum which is not direct: $V+V=V$ for all vector spaces $V$. Any sum of two vectors in $V$ is again a vector in $V$, from which the equality follows. This sum is only direct if $V=0$. Otherwise there are obviously many ways to express a vector in $V$ as a sum of two other vectors, also from $V$. A less trivial example: Take a 3d vector space $V$ with basis $\{b_1,b_2,b_3\}$, and the subspaces $U_1$ spanned by $\{b_1,b_2\}$ and $U_2$ spanned by $\{b_2,b_3\}$. Then every vector in $V$ can be written as a sum of one vector from $U_1$ and one from $U_2$, but not uniquely so. For instance, $b_2$ can be written as $2b_2-b_2$, or as $b_2+0$, or many other combinations.