Linear Algebra - Vectors

Vectors

Definition

A vector in $\mathbb{R}^n$ is an ordered n-tuple $(v_1, v_2, \dots, v_n)$.

Vector Operations

Addition: $v + w = (v_1 + w_1, v_2 + w_2, \dots, v_n + w_n)$

Scalar Multiplication: $cv = (cv_1, cv_2, \dots, cv_n)$

Properties

• Commutative: $v + w = w + v$
• Associative: $(u + v) + w = u + (v + w)$
• Distributive: $c(v + w) = cv + cw$
• Additive identity: $0$ vector
• Multiplicative identity: $1 \cdot v = v$

Dot Product (Inner Product)

Definition: $v \cdot w = v_1w_1 + v_2w_2 + \dots + v_nw_n$

Properties:

• $v \cdot w = w \cdot v$ (commutative)
• $(u + v) \cdot w = u \cdot w + v \cdot w$
• $(cv) \cdot w = c(v \cdot w)$

Geometric Meaning: $v \cdot w = |v||w|\cos \theta$, where $\theta$ is angle between vectors.

Norm (Magnitude)

$|v| = \sqrt{v \cdot v} = \sqrt{v_1^2 + v_2^2 + \dots + v_n^2}$

Properties:

• $|v| \geq 0$, and $|v| = 0 \iff v = 0$
• $|cv| = |c||v|$
• Triangle inequality: $|v + w| \leq |v| + |w|$
• Cauchy-Schwarz: $|v \cdot w| \leq |v||w|$

Unit Vectors

$u = v/|v|$ is unit vector in direction of $v$.

Standard basis: $i = (1,0,0), j = (0,1,0), k = (0,0,1)$ in $\mathbb{R}^3$

Cross Product (\mathbb{R}^3 only)

$v \times w = (v_2w_3 - v_3w_2, v_3w_1 - v_1w_3, v_1w_2 - v_2w_1)$

Properties:

• $v \times w = -(w \times v)$ (anti-commutative)
• $v \times (w + u) = v \times w + v \times u$
• $(cv) \times w = c(v \times w)$

Geometric Meaning:

• $|v \times w| = |v||w|\sin \theta$ (area of parallelogram)
• $v \times w$ is perpendicular to both $v$ and $w$
• Right-hand rule determines direction

Scalar Triple Product (\mathbb{R}^3)

$u \cdot (v \times w)$ gives volume of parallelepiped.

Property: $u \cdot (v \times w) = (u \times v) \cdot w = (w \times u) \cdot v$