Condition for Direct Sums

The definition of direct sum requires every vector in the sum to have a unique representation as an appropriate sum. The next result shows that when deciding whether a sum of subspaces is a direct sum, we only need to consider whether $0$ can be uniquely written as an appropriate sum.

Condition for Direct Sums

Suppose $V_1,\dots ,V_m$ are subspaces of $V$. Then, $V_1+\cdots +V_m$ is a direct sum if and only if the only way to write $0$ as a sum $v_1+\cdots +v_m$, where each $v_k\in V_k$, is by taking each $v_k$ equal to $0$.

Proof.

First, we show necessity: assuming that $V_1+\cdots +V_m$ is a direct sum, the only unique way to write the zero vector is if each $v_k=0$.

Suppose $V_1+\cdots +V_m$ is a direct sum. All of the subspaces $V_1,\dots ,V_m$ are subspaces of $V$, which means all of these subspaces share the same additive identity $0$. Clearly, adding all of these additive identities $0_1+\cdots +0_m$, where each $0_k\in V_k$, is equal to $0$. Then, the definition of direct sum implies that this is the only way to write $0$, since this $0$ must unique.

Thus, the definition of direct sum implies that the only way to write $0$ as a sum $v_1+\cdots +v_m$, where each $v_k\in V_k$, is by taking each $v_k$ equal to $0$.

Second, we show sufficiency: assuming that the only way to express the zero vectors is by having all $v_k=0$, this implies that $V_1+\cdots +V_m$ is a direct sum.

Suppose that the only way to write $0$ as a sum $v_1+\cdots +v_k$, where $v_k\in V_k$, is by taking each $v_k$ equal to $0$.

To show that $V_1+\cdots +V_m$ is a direct sum, let some $v\in V_1+\cdots +V_k$. We can write

$$v=v_1+\cdots +v_m$$

for some $v_1\in V_1,\dots ,v_m\in V_m$. To show that this representation is unique, suppose we also have

$$v=u_1+\cdots +u_m,$$

where $u_1\in V_1,\dots ,u_m\in V_m$. Subtracting these two equations, we have:

$$0=(v_1-u_1)+\cdots +(v_m-u_m).$$

Because $v_1-u_1\in V_1,\dots ,v_m-u_m\in V_m$, the equation above implies that each $v_k-u_k$ equals $0$. This shows that any two representations of the same vector must be equal, so every vector actually only has one unique representation.