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.