[Finite-Dimensional Linear Algebra] 2.2 Vector Spaces + 2.3 Subspaces

The elements of vector space $V$ are called vectors, and the elements of the corresponding field $F$ are called scalars. 

Definition of a Vector Space

Let $F$ be a field and let $V$ be a nonempty set with two operations defined with respect to these sets: 

• (vector) addition: $u, v \in V \Rightarrow  u + v \in V$
• scalar multiplication: $\alpha \in F, v \in V \Rightarrow \alpha v \in V$

 

$V$ is a vector space over $F$ iff the following properties are satisfied: 

A. Axioms for Addition

1. Associative property of addition
   • for all $u, v, w \in V$,  $$u + (v+w) = (u+v) + w$$
2. Commutative property of addition
   • for all $u, v \in V$,  $$u+v = v+u$$
3. Existence of an additive identity
   • there exists a unique element 0 of $V$ for all $u \in V$ such that  $$u + 0 = u$$
4. Existence of additive inverses
   • for each $u \in V$, there exists a unique element $-u$ of $V$ such that $$u + (-u) = 0$$

B. Axioms for Scalar Multiplication

1. Associative property of multiplication
   • for all $\alpha , \beta \in F, u \in V$,  $$\alpha (\beta u)= (\alpha \beta )u$$
2. Multiplicative identity 1 of $F$ (!) 
   • for all $u \in V$ where 1 is the multiplicative identity of $F$, $$u \cdot 1 = u$$

C. Distributive Axioms

1. Distributive property of multiplication over addition
   • for all $\alpha \in F, u, v \in V$,  $$\alpha (u + v) = \alpha u + \alpha v$$
2. Distributive property of addition over multiplication (!)
   • for all $\alpha , \beta \in F, u \in V$,  $$(\alpha + \beta )u = \alpha u + \beta u$$

Q. How are vector space axioms different from those of a field?
A. Axioms that differ from or are not common to the definition of fields were marked with (!). Also note that the commutative property of multiplication and the existence of multiplicative inverses are NOT properties of vector spaces.
1. Commutative property of multiplication: for all $\alpha , \beta \in F$, $\alpha \beta = \beta \alpha$
2. Existence of multiplicative inverses: for each $\alpha \in F$, $\alpha \neq 0$ there exists an element $\alpha^{-1} \in F$ with the property that $\alpha \alpha^{-1} = 1$

Euclidean n-space

Real Euclidean n-space $\mathbb{R}^{n}$ is an n-dimensional vector space over the field $\mathbb{R}$, defined to be a collection of all n-tuples of real numbers:

\mathbb{R}^{n} = $$\left\{\begin{array}{c}\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]:x_1,x_2,\cdots ,x_n\in {\mathbb{R}}^n\end{array}\right\}$$

Given $x \in \mathbb{R}^{n}$, the numbers $x_{1}, x_{2}, \cdots , x_{n}$ are called components of $x$, and $x$ can be written in the alternate notation $$x = (x_{1}, x_{2}, \cdots , x_{n})$$

• addition: for $x, y \in \mathbb{R}^{n},$ $$x+y=\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]+\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_n\end{array}\right]=\left[\begin{array}{c}{x}_1+y_1\\ x_2+y_2\\ ⋮\\ x_n+y_n\end{array}\right]$$
• scalar multiplication: for $\alpha \in \mathbb{R},x\in {\mathbb{R}}^n,$ $$\alpha x=\alpha \left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]=\left[\begin{array}{c}\alpha x_1\\ \alpha x_2\\ ⋮\\ \alpha x_n\end{array}\right]$$

Complex Euclidean n-space $\mathbb{C}^{n}$, an n-dimensional vector space over the field $\mathbb{C}$, is analogous to $\mathbb{R}^{n}$.

In general, given any field $F$, the space $F^{n}$ can be defined as vector space over $F$.

• Taking $n=1$ would be equivalent to regarding field $F$ as vector space over itself.

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Definition of a Subspace

Let S be a subset of $V$, which is a vector space over a field $F$. Then $S$ is a subspace of $V$ iff the following are true:

1. Contains the additive identity $$0 \in S$$
2. Closed under scalar multiplication
   • if $\alpha \in F$ and $u \in S$, then $$\alpha u \in S$$
3. Closed under addition
   • if $u, v \in S$, then $$u+v \in S$$

Figure 2.3.1. Vectors $\mathbf{u} +  \mathbf{v}$ and $k \mathbf{u}$ lie in the same plane as $\mathbf{u}$ and $\mathbf{v}$.

In Figure 2.3.1, the plane $W$ passes through the origin and is closed under addition and scalar multiplication. Thus, $W$ is a subspace of $R^3$ 

 

Given any vector space $V$, the set containing only the zero vector, $S={0}$, is a subspace of $V$ , and is called the trivial subspace. The entire space $V$  is a subspace of itself as well. Any subspace of $V$ that is neither trivial nor is all of $V$ is called a proper subspace of $V$.

• If $U$ is a proper subspace of an n-dimensional vector space $V$, then $U$ can be m-dimensional where $m<n$.

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vector: element of a vector space

scalar: element of a field

vector space: abstract space that encompasses ordinary Euclidean vectors and functions; defined by two linear operations of addition and scalar multiplication

subspace: a subset of vector space that is itself a vector space

trivial subspace: a set containing only the zero vector

proper subspace: a set containing nonzero vectors other than the vector space itself