Linear Algebra - Vector Spaces

Vector Spaces

Definition

Set V with operations + (addition) and · (scalar multiplication) satisfying:

1. $u + v = v + u$ (commutativity)
2. $(u + v) + w = u + (v + w)$ (associativity)
3. Exists zero vector 0 with $v + 0 = v$
4. Exists additive inverse $-v$ with $v + (-v) = 0$
5. $1 \cdot v = v$
6. $c(dv) = (cd)v$ (associativity of scalar multiplication)
7. $c(u + v) = cu + cv$ (distributivity)
8. $(c + d)v = cv + dv$ (distributivity)

Examples:

• $\mathbb{R}^n$: n-tuples of real numbers
• $M_{m \times n}$: $m \times n$ matrices
• $P_n$: polynomials of degree $\leq n$
• $C[a,b]$: continuous functions on $[a,b]$

Subspaces

Subset $W$ of $V$ is subspace if closed under + and scalar multiplication (contains zero vector).

Test: For all $u,v$ in $W$ and scalars $c$: $u+v$ in $W$, $cu$ in $W$.

Span

$\quad ext{Span}\{v_1, \dots, v_k\} = \{c_1v_1 + \dots + c_kv_k : c_i \text{ scalars}\}$

Collection of all linear combinations.

Basis and Dimension

Basis: Linearly independent set that spans the space.

Standard basis for $\mathbb{R}^n$: $\{e_1, e_2, \dots, e_n\}$ where $e_i$ has 1 in i-th position.

Dimension: Number of vectors in basis.

All bases have same size.

Basis Theorem: Any linearly independent set can be extended to basis.

Any spanning set can be reduced to basis.

Coordinate Vectors

Given basis $B = \{v_1, \dots, v_n\}$, any $v = c_1v_1 + \dots + c_nv_n$.

$[v]_B = (c_1, \dots, c_n)$ is coordinate vector relative to B.

Row Space and Column Space

Row space: Span of rows of matrix.

Column space: Span of columns of matrix.

Fundamental Theorem of Linear Algebra:

• $\text{rank}(A) = \text{dimension of column space} = \text{dimension of row space}$
• $\text{rank}(A) + \text{nullity}(A) = n$

Null Space

Null space: $N(A) = \{x : Ax = 0\}$

Dimension = $\text{nullity}(A) = \text{number of free variables}$

Basis: Use special solutions from RREF.

Rank-Nullity Theorem

For $m \times n$ matrix A:

• $\text{rank}(A) + \text{nullity}(A) = n$
• $\text{rank}(A) \leq \min(m, n)$