Direct Sums of Subspaces

Suppose $V_1,\dots ,V_m$ are subspaces of $V$. Every element of the sums of subspaces $V_1+\cdots +V_m$ can be written in the form

$$v_1+\cdots +v_m$$

where each element $v_k\in V_k$.

The sum $V_1+\cdots +V_m$ is called the direct sum if each element on $V_1+\cdots +V_m$ can be written in only one way as a sum $v_1+\cdots +v_m$, where each $v_k\in V_k$.

If $V_1+\cdots +V_m$ is a direct sum, then $V_1\oplus \cdots \oplus V_m$ denotes $V_1+\cdots +V_m$, with the $\oplus$ notation serving as an indication that this is a direct sum.

Sums of subspaces are analogous to unions of subsets. Similarly, direct sums of subspaces are analogous to disjoint unions of subsets. No two subspaces of a vector space can be disjoint, because both contain $0$. So disjointness is replaced, at least in the case of two subspaces, with the requirement that the intersection equal $\{0\}$.

Basic Examples

Example: Direct sum of two subspaces

Suppose $U$ is the subspace of ${\mathbb{F}}^3$ of vectors whose last coordinate equals $0$, and $W$ is the subspace of ${\mathbb{F}}^3$ of vectors whose first two coordinates equal $0$:

$$\begin{array}{rl}U& =\{(x,y,0)\in {\mathbb{F}}^3:x,y\in \mathbb{F}\}\\ W& =\{(0,0,z)\in {\mathbb{F}}^3:z\in \mathbb{F}\}\end{array}$$

Then, ${\mathbb{F}}^3=U\oplus W$.

Example: A non-direct sum

Suppose

$$\begin{array}{rl}{V}_1& =\{(x,y,0)\in {\mathbb{F}}^3:x,y\in \mathbb{F}\},\\ V_2& =\{(0,0,z)\in {\mathbb{F}}^3:z\in \mathbb{F}\},\\ V_3& =\{(0,y,y)\in {\mathbb{F}}^3:y\in \mathbb{F}\}\end{array}$$

Then ${\mathbb{F}}^3=V_1+V_2+V_3$ because every vector $(x,y,z)\in {\mathbb{F}}^3$ can be written as

$$(x,y,z)=(x,y,0)+(0,0,z)+(0,0,0)$$

where the first vector on the right side is in $V_1$, the second vector is in $V_2$, and the third vector is in $V_3$.

However, ${\mathbb{F}}^3$ is not a direct sum of $V_1,V_2,V_3$, because the vector $(0,0,0)$ can be written in more than one way as a sum $v_1+v_2+v_3$, with each $v_k\in V_k$. Specifically, we have:

$$(0,0,0)=(0,1,0)+(0,0,1)+(0,-1,-1)$$

and

$$(0,0,0)=(0,0,0)+(0,0,0)+(0,0,0)$$

where the first vector is in $V_1$, the second vector is in $V_2$, and the third vector is in $V_3$.

Thus, the sum $V_1+V_2+V_3$ is not a direct sum.