5.4 Direct sums

Another useful concept related to subspaces is the notion of a direct sum, which is a stronger condition than the sum defined in Theorem 5.28.

Definition 5.31: (Direct sum)

A subspace $V\subseteq W$ is said to be the direct sum of two subspaces $V_1,V_2\subseteq W$, with the notation $V=V_1\oplus V_2$ if

• $V=V_1+V_2$, and

• $V_1\cap V_2=\{\mathbf{0}\}$.

Example 5.32:

Consider $V_1=\operatorname{span}\{e_1\}$, $V_2=\operatorname{span}\{e_2\}$ with $e_1=(1,0), e_2=(0,1)\in\mathbb{R}^2$, then $$\mathbb{R}^2=V_1\oplus V_2 .$$

If a subspace $V$ is the sum of two subspaces $V_1, V_2$, every element of $V$ can be written as a sum of two elements of $V_1$ and $V_2$, and if $V$ is a direct sum this decomposition is unique.

Theorem 5.33:

Let $V_1$ and $V_2$ be subspaces of a vector space. If $W=V_1\oplus V_2$, then for any $w\in W$ there exist unique $v_1\in V_1,v_2\in V_2$ such that $w=v_1+v_2$.

Proof.

It is clear that there exist $v_1,v_2$ with $w=v_1+v_2$ from the sum part of the direct sum definition. So what we have to show is uniqueness. So Let us assume there is another pair $v_1'\in V_1$ and $v_2'\in V_2$ such that $w=v_1'+v_2'$, then we can subtract the two different expressions for $w$ and obtain $$\mathbf{0}=(v_1+v_2)-(v_1'+v_2')=v_1-v_1'-(v_2'-v_2)$$ and therefore $v_1-v_1'=v_2'-v_2$. But in this last equation the left hand side is a vector in $V_1$, the right hand side is a vector in $V_2$ and since they have to be equal, they lie in $V_1\cap V_2=\{\mathbf{0}\}$, so $v_1'=v_1$ and $v_2'=v_2$.

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A trivial but useful example is as follows.

Example 5.34:

Consider the subspaces $$V_1=\{(x,y,0): x,y\in \mathbb{R}\}$$ and $$V_2= \{(0,0,z): z,\in \mathbb{R}\}$$ of $\mathbb{R}^3.$ Then for any $v\in \mathbb{R}^3,$ there is a unique decomposition $$v=(x,y,z) = (x,y,0,)+(0,0,z)$$ as a sum of two elements from $V_1$ and $V_2$. In other words, we can think of $\mathbb{R}^3$ as the direct sum of the $xy$-plane and the $z$-axis.

Exercise 5.35:

Let $V_1=\{(x_1,x_2,x_3) \in \mathbb{R}^3: x_1+x_2=x_3\}$ and $V_2=\{(y_1, y_2, y_3)\in \mathbb{R}^3: y_1-y_2=y_3\}$. Does $\mathbb{R}^3=V \oplus W$?

Once again, we may find ourselves in a situation where we have a system of linear equations to solve when trying to determine whether we have a direct sum, and the techiniques from Chapter 3 come in useful.

While exploring the idea of subspace, we have seen that it is possible to take a set of vectors and use it to construct a subspace. A natural question is how can we do this in the most `efficient’ way, that is using the smallest number of vectors possible. This is what we will explore in the next chapter.