Linear algebra extends its foundation into vector spaces, a core concept that generalizes vectors beyond simple geometric interpretations.

Key Topics

  • Definition of Vector Spaces: A set closed under addition and scalar multiplication, with axioms like associativity and distributivity.
  • Subspaces: Non-empty subsets that satisfy vector space properties under inherited operations.
  • Span and Linear Independence: Span of vectors forms a subspace; linearly independent sets generate bases.
  • Basis and Dimension: A minimal spanning set (basis) determines the dimension of a space.
vector_space

Example:

Consider vectors in ℝ³. The set of all linear combinations of two non-parallel vectors forms a plane (2D subspace).

linear_transformation

For deeper exploration, check Chapter 5: Linear Transformations to see how vector spaces connect to function mappings.

basis_vectors

Practice Problems

  1. Prove that the span of a single vector is a subspace.
  2. Find a basis for the solution space of the system:
    $$ \begin{cases} x + y = 0 \ 2x + 2y = 0 \end{cases} $$

Expand your understanding with this resource on foundational concepts.