## 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:**

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 \(w=v_1+v_2\).

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:**

**Exercise 5.35:**

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.