Section 4.2 Vector Subspaces
Definition 4.2.1. Vector subspace.
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 \(V\text{.}\)
Example 4.2.2.
If \(V\) is a vector space over \(\R\text{,}\) then \(\{0\}\) and \(V\) are two trivial subspaces of \(V\text{.}\)
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.
Theorem 4.2.3.
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 \(V\text{.}\)
Example 4.2.4.
Let \(V=\R^2\) with usual addition and scalar multiplication.
(i) Any line in \(\R^2\) passing through the origin is a subspace of \(\R^2\text{.}\)
(ii) The line \(L=\{(t,1-t)\}\) is not a subspace of \(\R^2\text{,}\) as it does not contain the origin.
(iii) 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.
(iv) 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.
In fact, \(\{0\}, \R^2\) and any line passing through origin are only subspaces of \(\R^2\text{.}\)
Example 4.2.5.
Let \(V=\R^3\) with usual addition and scalar multiplication.
(i) Any line in \(\R^3\) passing through the origin is a subspace of \(\R^3\text{.}\)
(ii) Any plane in \(\R^3\) passing through the origin is a subspace of \(\R^3\text{.}\)
(iii) 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.
Example 4.2.6.
Let \(A\) be an \(m\times n\) real matrix. Then we have the following subspaces associated to \(A\text{.}\)
(i) \({ Null}(A)={\cal N}(A)={ ker}(A)\) is subspace of \(\R^n\)
(ii) \({Row}(A)\) is subspace of \(\R^n\)
(iii) \({ Col}(A)={ Im}(A)={\cal R}(A)\) is subspace of \(\R^m\)
(iv) \({\cal L}(A)={ Null}(A^T)\) is subspace of \(\R^m\text{.}\)
The above four subspaces are called fundamental subspaces associated to \(A\text{.}\)
Example 4.2.7.
Let \(V = M_n(\R)\text{,}\) the set of all \(n\times n\) real matrices with usual matrix addition and scalar multiplication.
(i) \(S=\{A\in M_n(\R): A=A^T\}\) is a subspace of \(V\text{.}\)
(ii) \(K=\{A\in M_n(\R): A+A^T=0\}\) is a subspace of \(V\text{.}\)
(iii) \(W=\{A\in M_n(\R): { trace}(A)=0\}\) is a subspace of \(V\text{.}\)
(iv) \(G_n(\R)\) is not a subspace of \(V\text{.}\)
(v) \(\{A\in V: { det}(A)=0\}\) is not a subspace of \(V\text{.}\)
Problem 4.2.8.
Fix a matrix \(P\in M_n(\R)\text{.}\) Define \(W=\{A\in M_n(\R):AP=PA\}\text{.}\) Show that \(W\) is a subspace of \(M_n(\R)\text{.}\)
Example 4.2.9.
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 4.1.7. Let us look at some of the subspaces of
\(V\text{.}\)
(i) \(B(\R)\text{,}\) the set of all bounded functions from \(\R\) to \(\R\) is a subspace of \(V\text{.}\)
(ii) \({\cal C}(\R)\text{,}\) the set of all continuous functions from \(\R\) to \(\R\) is a subspace of \(V\text{.}\)
(iii) \({\cal D}(\R)\text{,}\) the set of all differentiable functions from \(\R\) to \(\R\) is a subspace of \(V\text{.}\)
(iv) 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.)
(iv) \(W\) is the set of even functions from \(\R\to \R\) is a subspace of \(V\text{.}\) What about set of odd functions?
Example 4.2.10.
Let \(V\) be a vector space over \(\R\text{.}\) Let \(W_1\) and \(W_2\) be two subspaces of \(V\text{.}\) Then
(i) \(W_1\cap W_2\) is a subspace of \(V\text{.}\) What about \(W_1\cup W_2\text{?}\)
(ii) \(W_1+W_2=\{w_1+w_2:w_1\in W_2,w_2\in W_2\}\) is a subspace of \(V\text{.}\)
Example 4.2.11.
Let \(V\) be a vector space over \(\R\text{.}\) Let \(S=\{v_1,\ldots, v_k\}\) be a subset of \(V\text{.}\) Then the linear span defined as
\begin{equation*}
L(S):=\{\alpha_1 v_1+\cdots+\alpha_kv_k:\alpha_1,\ldots,\alpha_k\in \R\}
\end{equation*}
is a subspace of \(V\text{.}\)
