4) Vector Spaces

Vector Spaces with Real-World Context

Vector Spaces with Real-World Context

1. Definition of a Vector Space

A vector space is a set of vectors that can be added together and multiplied by scalars, following specific rules (axioms). These properties allow vectors to be manipulated in a consistent way.

Real-World Example:

  • Geometric Space: Any position or direction in a 2D or 3D world can be represented as a vector.
  • Function Spaces: The set of all polynomials with real coefficients forms a vector space.

2. Properties of a Vector Space (Axioms)

Addition Properties:

u + v = v + u (Commutativity)

(u + v) + w = u + (v + w) (Associativity)

v + 0 = v (Zero Vector Exists)

v + (-v) = 0 (Additive Inverse Exists)

Scalar Multiplication Properties:

c (d v) = (cd) v (Associativity of Scalar Multiplication)

c (u + v) = c u + c v (Distributivity of Scalar over Vector Addition)

(c + d) v = c v + d v (Distributivity of Scalar Addition)

1 v = v (Multiplicative Identity)

3. Subspaces

A subspace of a vector space is a subset that is also a vector space under the same operations.

Real-World Example:

  • A flat plane within a 3D space (like a tabletop) is a subspace of the larger 3D space.

Conditions for a Subspace:

  • Contains the zero vector.
  • Closed under addition.
  • Closed under scalar multiplication.

4. Basis and Dimension

A basis is a set of linearly independent vectors that span the space.

The dimension of a vector space is the number of vectors in its basis.

Real-World Example:

  • For 2D space, the basis is {(1,0), (0,1)}.
  • In 3D, the basis is {(1,0,0), (0,1,0), (0,0,1)}.

5. Coordinate Systems and Change of Basis

Coordinate systems allow us to express vectors in terms of basis vectors. Changing the basis lets us transform vectors into different coordinate systems.

Real-World Example:

  • In computer graphics, changing the basis is used to render objects from different viewpoints.

Summary of Real-World Meanings

  • Vector Spaces allow consistent manipulation of vectors.
  • Subspaces are smaller vector spaces within larger ones.
  • Basis and Dimension define the minimal set of vectors needed to span a space.
  • Change of Basis is fundamental in physics, engineering, and graphics.

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