Vector Spaces Review
Vector Spaces Review
We will now summarize some of the material mentioned on earlier pages regarding vector spaces.
- Recall that a Vector Space over the field $\mathbb{F}$ is a nonempty set with two operations of addition and scalar multiplication using the scalars from $\mathbb{F}$. If $V$ is a vector space, then for all $u, v, w \in V$ and $a, b \in \mathbb{F}$, the following ten axioms must be satisfied:
| 1. Commutativity of Addition | 2. Associativity of Addition | 3. Existence of a Zero Vector | 4. Existence of Additive Inverses |
|---|---|---|---|
| $u + v = v + u$ | $u (v + w) = (u + v) + w$ | $u + 0 = u = 0 + u$ | $u + (-u) = 0 = (-u) + u$ |
| 5. Associativity of Scalar Mult. | 6. Existence of Multiplicative Identity | 7. Distributivity over Vector Addition | 8. Distributivity over Scalar Mult. |
| $a(bu) = (ab)u$ | $1u = u$ | $a(u + v) = au + av$ | $(a + b)u = au + bu$ |
| 9. Closure under Addition | 10. Closure under Scalar Multiplication | ||
| $u, v \in V$ implies $(u + v) \in V$. | $a \in \mathbb{F}$ and $u \in V$ implies $au \in V$. |
- Some examples of vector spaces (with the obvious definitions for addition and scalar multiplication) are: $0$ (the zero vector space which contains only the zero vector), $\mathbb{F}^n$ (the set of all n-component vectors whose components are in $\mathbb{F}$, $M_{mn}$ (the set of all m by n matrices whose entries are in $\mathbb{F}$.
- Some further examples of vector spaces are $\wp_n (\mathbb{F})$ which is the vector space of all polynomials of degree less than or equal to $n$ and whose coefficients are in $\mathbb{F}$, $\mathbb{F}^{\infty}$ which is the set of all infinite sequences whose terms are in $\mathbb{F}$, or even $F(-\infty, \infty)$ which is the set of all real-valued functions. The definitions of addition and scalar multiplication defined on these vector spaces are once again the obvious ones.
- Vector spaces satisfy many properties which can be derived by the ten axioms listed above. For example, if $V$ is a vector space then for each vector $x \in V$ there exists a unique additive inverse $-x$. To show this, suppose that two additive inverses exist, say $-x$ and $-x'$. Then we have that:
\begin{equation} -x = -x + 0 = -x + (x -x') = (-x + x) - x' = 0 - x' = x' \end{equation}
- Recall that a Vector Subspace, say $U$ of the vector space $V$ is a subset $U \subseteq V$ that is itself a vector space under the addition and scalar multiplication defined on $V$. Note that the zero space $0 = \{ 0 \}$ is a subspace of every vector space, and the whole space $V$ is also a subspace of every vector space $V$.
- For a more complicated example - the subspaces of $\mathbb{R}^2$ are the zero vector space, the whole space $\mathbb{R}^2$, and all lines that pass through the origin. The subspaces of $\mathbb{R}^3$ are the zero vector space, the whole space $\mathbb{R}^3$, all the lines that pass through the origin, and all of the planes that pass through the origin.
- To check if a subset $U$ of $V$ is a subspace, we need to verify only axioms 9 and 10 above since the other axioms are inherited from the fact that $U \subseteq V$.
- Also, recall the important lemma from the Vector Subspaces page that if $U$ is a subset of the $\mathbb{F}$-vector space $V$, then to verify $U$ is a vector space, all we need to show is that if $x, y \in U$ and $a, b \in \mathbb{F}$ then $(ax + by) \in U$.
- Now also recall from the Vector Subspace Sums page that if $U_1$, $U_2$, …, $U_m$ are all vector spaces of $V$, then we can form the following Subspace Sum which is the defined to be the set of all vectors $v \in V$ that can be written as $v = u_1 + u_2 + ... + u_m$ where $u_i \in U_i$ for each $i = 1, 2, ..., m$:
\begin{align} \quad \sum_{i=1}^{m} U_i = U_1 + U_2 + ... + U_m \end{align}
- We say that the vector space $V$ is equal to the sum of subspaces $U_1$, $U_2$, …, $U_m$ if every vector $v \in V$ can be written as this sum.
- Recall from the Vector Sum Theorems page that we proved that if $U_1$ and $U_2$ are subspaces of $V$ then $U_1 + U_2$ is a subspace of $V$. We also say that $U_1 + U_2$ is the smallest subspace of $V$ that contains the subspaces $U_1$ and $U_2$.
- Furthermore, we say that $V$ is a Direct Sum of the subspaces $U_1$, $U_2$, …, $U_m$ if $V$ is equal to the sum of these subspaces and if every vector $v \in V$ can be written uniquely as the sum $v = u_1 + u_2 + ... + u_m$ where $u_i \in U_i$ for each $i = 1, 2, ..., m$. We denote this by:
\begin{align} \quad V = \bigoplus_{i=1}^{m} = U_1 \oplus U_2 \oplus ... \oplus U_m \end{align}
- Recall from the Direct Sum Theorems page that we proved that if $U_1$ and $U_2$ are two vector spaces of $V$, then $V = U_1 \oplus U_2$ if and only if $U_1 \cap U_2 = \{ 0 \}$. Note that this theorem does NOT necessarily hold in forming direct sums from more than two subspaces.