Vector Spaces

Chapter 6 • Section 6-1

"Vectors are everywhere... polynomials, matrices, functions! 🤯"

🌌 What is a Vector Space?

A vector space $V$ is a set of objects (called vectors) with two operations: Addition and Scalar Multiplication, satisfying 10 axioms.

Addition Axioms

A1. Closure: $\vec{u} + \vec{v} \in V$
A2. Commutativity: $\vec{u} + \vec{v} = \vec{v} + \vec{u}$
A3. Associativity: $(\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w})$
A4. Zero Vector: $\exists \vec{0} \in V$ s.t. $\vec{u} + \vec{0} = \vec{u}$
A5. Additive Inverse: $\exists -\vec{u} \in V$ s.t. $\vec{u} + (-\vec{u}) = \vec{0}$

Scalar Multiplication Axioms

S1. Closure: $c\vec{u} \in V$
S2. Distributive (Vector): $c(\vec{u} + \vec{v}) = c\vec{u} + c\vec{v}$
S3. Distributive (Scalar): $(c+d)\vec{u} = c\vec{u} + d\vec{u}$
S4. Associativity: $c(d\vec{u}) = (cd)\vec{u}$
S5. Identity: $1\vec{u} = \vec{u}$

Common Vector Spaces

🔢

$\mathbb{R}^n$

The classic space of column vectors with $n$ entries.

📈

Polynomials $\calP_n$

Polynomials of degree at most $n$. e.g., $1 + 2x - x^2$.

▦

Matrices $\M_{mn}$

The set of all $m \times n$ matrices.

📈

Functions $C[a,b]$

Continuous functions on interval $[a,b]$.

✨ Basic Properties

From the axioms, we can prove several useful properties that hold in every vector space:

  • Cancellation: If $\vec{u} + \vec{v} = \vec{u} + \vec{w}$, then $\vec{v} = \vec{w}$.
  • Zero Scaling: $0\vec{v} = \vec{0}$ for any vector $\vec{v}$.
  • Scaling Zero: $c\vec{0} = \vec{0}$ for any scalar $c$.
  • Negation: $(-1)\vec{v} = -\vec{v}$.

Polynomial Arithmetic Visualizer

See how adding polynomials works just like adding vectors component-wise.

$x^2 +$ $x +$
+
$x^2 +$ $x +$

🧠 Knowledge Check Win $15

Which of the following is NOT a vector space?

Hint: Is it closed under addition? What is $x^2 + (-x^2)$?