In linear algebra, we often encounter a set of vectors that themselves possess the structure of a vector space, but live naturally inside a larger vector space. For example, the set of all solutions to a homogeneous system of linear equations in \(\mathbb{R}^n\) may not the whole space \(\mathbb{R}^n\text{,}\) yet it still satisfies the axioms of a vector space. Such subsets, closed under vector addition and scalar multiplication, are called subspaces.
Let \(V\) be a vector space over \(\R\) and \(W\) a nonempty subset of \(V\text{.}\) The \(W\) is called a subspace of \(V\text{,}\) if \(W\) itself is a vector space under the inherited vector addition and scalar multiplication on \(V\text{.}\)
Let \(V\) be a vector space over \(\R\) and \(W\) a nonempty subset of \(V\text{.}\) Assume that \(W\) is closed under addition and scalar multiplication on \(V\text{.}\) For \(W\) to be a subspace of \(V\text{,}\) we need to show that all the eight properties in definition of vector space must be satisfied for elements in \(W\text{.}\) Fortunately, our task is simplified as most of these properties are inherited from the vector space \(V\text{.}\) Let \(w\in W\text{,}\) the \(0\cdot w=0\in W\text{.}\) Note that we require \(W\neq \emptyset\) for this property. Also for \(w\in W\text{,}\)\((-1)w=-w\in W\text{.}\) Thus \(W\) contains the additive identity and additive inverse. Remaining conditions are true as elements in \(V\) and hence are also true as elements of \(W\text{.}\) This show that if \(W\) nonempty subset of \(V\) which is closed under addition and scalar multiplication, then it is subspace of \(V\text{.}\)
If \(W\) itself is a vector space under the addition and scalar multiplication on \(V\text{,}\) then \(W\) is closed under addition and scalar multiplication. Thus we have the following result.
Let \(V\) be a vector space over \(\R\) and \(W\) a nonempty subset of \(V\text{.}\) Then \(W\) is a subspace of \(V\) if and only if \(W\) is closed under addition and scalar multiplication.
Thus in order to check that if a nonempty subset \(W\subset V\) is a subspace, all we need to check that it is closed under addition and scalar multiplication inherited from \(V\text{.}\)
The \(W_1=\{(x,y):x\geq 0,y\geq 0\}\text{,}\) the first quadrant is not a subspace as it is not closed under scalar multiplication (why?). However, it is closed under addition.
The \(W_2=\{(x,y):xy\geq 0\}\text{,}\) the the union of first and third quadrant is not a subspace as it is not closed addition (why?). However it is closed under scalar multiplication.
If \(W\) is a subspace of \(\R^3\text{,}\) then \(W\) is one of the following: \(\{0\}\text{,}\)\(\R^3\text{,}\) a line passing through origin, a plane passing through origin.
Let \(V=\{f\colon \R\to \R\}={\cal F}(\R,\R)\) set of all functions from \(\R\to \R\) with addition and scalar multiplication defined as in Example 4.1.8. Let us look at some of the subspaces of \(V\text{.}\)
\(B(\R)\text{,}\) the set of all bounded functions from \(\R\) to \(\R\) is a subspace of \(V\text{.}\)
Fix \(a\in \R\) and \(W=\{f\colon \R\to \R: f(a)=0\}\) is a subspace of \(V\text{.}\) (What if we take all functions vanishing at finitely many points.)
