Vector Spaces

Math 303: Section 22

Dr. Janssen

Intro

\[ \def\R{{\mathbb R}} \def\P{{\mathbb P}} \def\b{{\mathbf{b}}} \def\x{{\mathbf{x}}} \def\y{{\mathbf{y}}} \def\u{{\mathbf{u}}} \def\v{{\mathbf{v}}} \def\w{{\mathbf{w}}} \def\z{{\mathbf{z}}} \DeclareMathOperator{\null}{Nul} \DeclareMathOperator{\span}{Span} \]

Preview Activity

Properties of addition and scalar multiplication in \(\P_1\), the space of linear polynomials.

Idea

Some mathematical structures behave like others!

Generally, abstraction is the process of deciding what structure/properties we care about. In linear algebra, we care about vector spaces.

The definition

A set \(V\) on which an operation of addition and a multiplication by scalars is defined is a vector space if for all \(\u,\v,\w\) in \(V\) and all scalars \(a\) and \(b\):

• \(\u + \v\) is an element of \(V\)
• \(\u + \v = \v + \u\)
• \((\u + \v) + \w = \u + (\v + \w)\)
• There is a zero vector \(\mathbf{0}\) in \(V\) so that \(\u + \mathbf{0} = \u\)
• For each \(\x\) in \(V\) there is a \(\y\) in \(V\) so that \(\x + \y = \mathbf{0}\)
• \(a\u\) is an element of \(V\)
• \((a+b)\u = a\u + b\u\)
• \(a(\u + \v) = a\u + b\v\)
• \((ab)\u = a(b\u)\)
• \(1\u = \u\)

we’ll focus on real scalars, but they can come from any field.

Note that this list of vector space axioms is meant to be as small as possible. There are things that are true of vector spaces which we do not include on the list because they can be proved using the axioms that are on the list.

A menagerie of examples

• \(\R^n\) with the usual vector addition and scalar multiplication
• \(\P_n\)¹
• The eigenspace of an \(n\times n\) matrix corresponding to an eigenvalue \(\lambda\) is a subspace of \(\R^n\), and therefore a vector space.
• The null space of a matrix
• The column space of a matrix
• The span of any set of vectors in \(\R^n\)
• The space \(\mathcal{M}_{m\times n}(\R)\) of \(m\times n\) matrices with real entries.
• The space \(\mathcal{F}\) of functions from \(\R\) to \(\R\)
• The space \(\R^\infty\) of infinite real sequences!
• The set of solutions to a second order homogeneous differential equation

The space \(\mathcal{M}_{m\times n}(\R)\) is the same as \(\R^p\) for what value of \(p\)?

1. Observe that \(\P_n\) is essentially the same as \(\R^{n+1}\)

Sample Theorem

Theorem 1 (Multiplying by 0) Let \(\v\) be a vector in a vector space \(V\). Then \(0\v = \mathbf{0}\).

standard proof: \(0\v = (0+0)\v\), etc.

Activity 22.1

Let \(\u,\v,\w\) be vectors in a vector space \(V\), and suppose that

\[ \u + \w = \v + \w \qquad(1)\]

• Why does \(V\) contain an additive inverse \(\mathbf{z}\) of \(\w\)?
• Add \(\z\) to both sides of Equation 1 to obtain

\[ (\u + \w) + \z = (\v + \w) + \z. \]

Which property of a vector space allows us to state the following?

\[ \u + (\w + \mathbf{z}) = \v + (\w + \mathbf{z}) \]

• Now use the properties of additive inverses and the additive identity to explain why \(\u = \v\). Conclude that we have a cancellation law for addition in any vector space.

Theorem 22.3

Let \(V\) be any vector space with identity \(\mathbf{0}\).

• \(0\v = \mathbf{0}\) for any \(\v\) in \(V\)
• The vector \(\mathbf{0}\) is unique.
• \(c\mathbf{0} = \mathbf{0}\) for any scalar \(c\).
• For any \(\v\) in \(V\), the additive inverse of \(\v\) is unique.
• The additive inverse of a vector \(\v\) in \(V\) is the vector \((-1)\v\).
• If \(\u, \v\), and \(\w\) are in \(V\) and \(\u + \w = \v + \w\), then \(\u = \v\).

Prove a couple of these, particularly one of the uniqueness statements.

Subspaces

Extending the idea

Often a vector space \(V\) contains a vector space \(W\). In this case, we say that \(W\) is a subspace of \(V\).

Activity 22.2

Let \(H = \{at : a\in \R\}\). Notice that \(H\) is a subset of \(\P_1\).

• Is \(H\) closed under the addition in \(\P_1\)? Verify.
• Does \(H\) contain the zero vector from \(\P_1\)? Verify.
• Is \(H\) closed under multiplication by scalars? Verify.
• Explain why \(H\) satisfies every other property of the definition of a vector space automatically just by being a subset of \(\P_1\) and using the same operations as in \(\P_1\). Conclude that \(H\) is a vector space.

Activity 22.2 illustrates an important point. There is a fundamental difference in the types of properties that define a vector space. Some of the properties that define a vector space are true for any subset of the vector space because they are properties of the operations (such as the commutative and associative properties). The other properties (closure, the inclusion of the zero vector, and the inclusion of additive inverses) are set properties, not properties of the operations. So these three properties have to be specifically checked to see if a subset of a vector space is also a vector space. This leads to the definition of a subspace, a subset of a vector space which is a vector space itself.

A consequence

Definition 1 (Subspace) A subset \(H\) of a vector space \(V\) is a subspace of \(V\) if

• whenever \(\u,\v\in H\) it is also true that \(\u+\v\in H\)
• whenever \(\u\) is in \(H\) and \(a\) is a scalar, it is also true that \(a\u\in H\)
• \(\mathbf{0}\in H\)

presented in the text as the definition, but in most texts as a theorem (a consequence of the definition)

Activity 22.3

Is the given subset \(H\) a subspace of the indicated vector space \(V\)? Verify your answer.

• \(V\) is any vector space and \(H = \{\mathbf{0}\}\)
• \(V = M_{2\times 2}\), and \(H = \left\{\left[\begin{matrix} 2x & y \\ 0 & x \end{matrix}\right] \ : \ x,y\in\R \right\}\)
• \(V = \P_2\), \(H = \{2at^2 + 1 \ : \ a\in\R\}\)
• \(V = \P_2\), \(H = \{at \ : \ a\in \R \} \cup \{bt^2 \ : \ b\in \R\}\)
• \(V = \mathcal{F}\), \(H = \P_2\)

interesting relationship in the polynomial spaces: \(\P_j \subseteq \P_k\) if and only if \(j\le k\); all are subsets of \(\P\), the space of all polynomials.

But a similar relationship does not hold for the reals: \(\R^2\not\subseteq \R^3\) (though it’s isomorphic to a subspace!)

Making subspaces from vectors

A given (randomly chosen?) set of vectors \(S\) is probably not a vector (sub)space. But we can build one by considering the span of \(S\).

From LC to Span

Definition 2 Let \(V\) be a vector space. A linear combination of vectors \(\v_1, \v_2, \ldots, \v_k\) in \(V\) is a vector of the form

\[ x_1 \v_1 + x_2 \v_2 + \cdots + x_k \v_k, \]

where \(x_1, x_2, \ldots, x_k\) are scalars. The span of the vectors \(\v_1, \v_2, \ldots, \v_k\) is the set of all linear combinations of \(\v_1, \v_2, \ldots, \v_k\):

\[ \span\{\v_1, \v_2, \ldots, \v_k\} = \{x_1 \v_1 + \cdots + x_k \v_k \ : \ x_i\in\R\}. \]

Spans are subspaces!

Activity 22.4

• Let \(H = \{ a_2 t^2 - a_1 t \ : \ a_1,a_2\in \R\}\). Note that \(H\) is a subset of \(\P_2\). Find two vectors \(\v_1,\v_2\in \P_2\) so that \(H = \span \{\v_1, \v_2\}\) and hence conclude that \(H\) is a subspace of \(\P_2\).

• Let \(p_1(t) = 1 - t^2\) and \(p_2(t) = 1 + t^2\), and let \(S = \{p_1(t), p_2(t)\}\) in \(\P_2\). Is the polynomial \(q(t) = 3 - 2t^2 \in \span S\)?

• With \(S\) as in the previous part, describe as best you can the subspace \(\span S\) of \(\P_2\).