Section 3.7 Subspaces
As
Investigation 3.6.5 shows, it can be very tedious to prove that a set is indeed a vector space. A
subspace of a vector space is a subset that is itself a vector space. Since most of the properties of the vector spaces we look at get inherited from some larger vector space, it is often easier to show that a set is a vector space by showing it is a subspace of the appropriate parent vector space.
Theorem 3.7.1.
A subset \(H\) of a vector space \(V\) is a subspace if and only if the following are true:
The zero vector of \(V\) is in \(H\text{;}\) \(\vec{0}_V \in H\text{.}\)
H is closed under vector addition; if \(\vec{u}, \vec{v} \in H\text{,}\) then \(\vec{u}+\vec{v}\in H\text{.}\)
H is closed under scalar multiplication; if \(\vec{u} \in H\) and \(c \in \mathbb{R}\text{,}\) then \(c\vec{u} \in H\text{.}\)
Proof.
We first assume that \(H\) is a subspace of \(V\text{.}\) Item (a) of the theorem follows from axiom 4 of being a vector space. Item (b) of the theorem follows from axiom 1 of being a vector space. Item (c) of the theorem follows from axiom 6 of being a vector space.
For the converse, we assume that \(H\subseteq V\) and the three items of the theorem satement are satisfied. We must verify the 10 vector space axioms:
This follows from item (b) of the theorem statement.
Since \(H\subseteq V\) and this axiom holds for all elements of \(V\text{,}\) the axiom holds when restricted to elements of \(H\text{.}\)
Since \(H\subseteq V\) and this axiom holds for all elements of \(V\text{,}\) the axiom holds when restricted to elements of \(H\text{.}\)
This follows from item (a) of the theorem statement.
This requires proof. Since \(-1\in\mathbb{R}\text{,}\) item (c) of the theorem statement tells us that for all \(\vec{u}\in H\text{,}\) \((-1)\vec{u}=-\vec{u}\in H\text{.}\) Thus, this axiom is verified.
This follows from item (c) of the theorem statement.
Since \(H\subseteq V\) and this axiom holds for all elements of \(V\text{,}\) the axiom holds when restricted to elements of \(H\text{.}\)
Since \(H\subseteq V\) and this axiom holds for all elements of \(V\text{,}\) the axiom holds when restricted to elements of \(H\text{.}\)
Since \(H\subseteq V\) and this axiom holds for all elements of \(V\text{,}\) the axiom holds when restricted to elements of \(H\text{.}\)
Since \(H\subseteq V\) and this axiom holds for all elements of \(V\text{,}\) the axiom holds when restricted to elements of \(H\text{.}\)
This theorem is so useful because we can prove a set is a vector space by checking only three properties instead of the ten that are involved in the definition. The reason that we do not need to check these other properties is that by using this subspace, we already have proven the proper rules of arithmetic from the parent space. Additionally, since we are using the same rules for scalar multiplication and vector addition as the parent space, we must also have the same scalars as the parent space.
Investigation 3.7.1.
Use the preceding theorem to prove that \(\mathbb{P}_n\) is a subspace of \(\mathbb{P}\text{.}\)
Investigation 3.7.2.
Is \(\mathbb{R}\) a subspace of \(\mathbb{C}\text{?}\) Why or why not?
Investigation 3.7.3.
Is \(\mathbb{R}^2\) a subspace of \(\mathbb{R}^3\text{?}\) Why or why not?
Investigation 3.7.4.
Is the set of points on the plane given by \(z=0\) a subspace of \(\mathbb{R}^3\text{?}\) Why or why not?
Investigation 3.7.5.
Is the set of points on the plane given by \(z=1\) a subspace of \(\mathbb{R}^3\text{?}\) Why or why not?
Investigation 3.7.6.
Draw a plot of the points in \(\mathbb{R}^2\) given by \(\{ \vec{x}=\colvec{x_1\\ x_2} \in
\mathbb{R}^2 \mid x_1 x_2 \geq 0\}\text{.}\) Is \(\{ \vec{x}=\colvec{x_1\\
x_2} \in \mathbb{R}^2 \mid x_1 x_2 \geq 0\}\) a subspace of \(\mathbb{R}^2\text{?}\) Why or why not?
Investigation 3.7.7.
Is \(Sym_{n \times n}\text{,}\) the set of symmetric \(n\) by \(n\) matrices a subspace of \(M_{n \times n}\text{?}\) Why or why not?
Investigation 3.7.8.
Prove or disprove: The set of odd functions on \(\mathbb{R}\) (i.e., those for which \(f(-t)=-f(t)\) for every \(t \in \mathbb{R}\)) a subspace of \(F(\mathbb{R},\mathbb{R})\text{.}\)
Solution.
The statement is true. We use Theorem 3.6 to prove this. First note that the function \(Z(t)=0\) is the zero vector for this vector space, as for any function \(f\colon \mathbb{R}\to\mathbb{R}\text{,}\) \((f+Z)(t) = f(t)+Z(t) =
f(t)+0 = f(t)\text{.}\) To see that \(Z\) is odd, we have \(Z(-t)
= 0=-0=-Z(t)\text{.}\) Now suppose that \(f,g\) are odd functions. We have
\begin{equation*}
(f+g)(-t) = f(-t)+g(-t)=-f(t)+(-g(t))=-(f(t)+g(t))=-(f+g)(t)\text{,}
\end{equation*}
verifying the second part of the theorem is satisfied. Finally, let \(c\in\mathbb{R}\text{.}\) Now \((cf)(-t) =
c(f(-t))=c(-f(t))=-cf(t)=-(cf)(t)\text{.}\) Thus, the set of odd functions are a subspace of the vector space of functions from \(\mathbb{R}\) to \(\mathbb{R}\text{.}\)
Theorem 3.7.2.
If \(A\) is a \(m\) by \(n\) matrix, the solution set to the homogeneous equation \(A\vec{x}=\vec{0}\) is a subspace of \(\mathbb{R}^n\text{.}\)
Theorem 3.7.3.
If \(H\) and \(K\) are subspaces of some vector space \(V\text{,}\) then the set \(H \cap K\) is a subspace of \(V\) as well.
Investigation 3.7.9.
Prove or Disprove: the set of \(2\) by \(2\) matrices with at least one zero entry is a subspace of \(M_{2 \times 2}\text{.}\)
Investigation 3.7.10.
Prove or Disprove: the set of matrices of the form \(\begin{bmatrix} a \amp b \\0 \amp -a \end{bmatrix}\) is a subspace of \(M_{2 \times 2}\text{.}\)
Investigation 3.7.11.
Prove or disprove: The set of quadratic polynomials of the form \(at^2+b\) is a subspace of the vector space of polynomials.
