Unit 1 of 10
Vector Spaces and Linear Maps
- 7 lessons
- Videos 1 to 7
This unit proves the basic facts about real and complex vector spaces and linear maps, including why every basis has the same size and the rank-nullity theorem.
It begins with how to read and write a proof, using quantifiers, converses, counterexamples and induction, and with what changes when complex scalars are allowed. It then builds the space of all linear maps between two spaces, proves rank-nullity by extending a basis of the kernel, and splits a space into a direct sum with a projection that need not be perpendicular.
Every lesson is one video. Each lesson page has the definitions and an example for every key term.
Proof and Scalar Fields
Proof Strategies in Linear Algebra
How to read and write proofs about vectors and linear maps, with universal claims, hypotheses and conclusions, converses, negating quantified statements, direct proof, counterexamples and induction.
Real and Complex Vector Spaces
Vector spaces over the real or complex numbers, with the complex conjugate and modulus, the vector space axioms, the subspace test, real versus complex dimension, and polynomials of degree at most n.
Video 2
Bases and Linear Maps
Basis Exchange and Dimension
Why every basis of a finite-dimensional space has the same length, proved with the exchange lemma, and how dimension, extracting and extending a basis, and the dimension of a subspace follow from it.
Video 3
The Vector Space of Linear Maps
Why a linear map is fixed by its basis images, how the linear maps between two spaces form a vector space under pointwise addition and scaling, why matrix-unit maps give it dimension mn, and what a rank-one map is.
Video 4
Rank-Nullity and Isomorphisms
The rank-nullity theorem and its proof by extending a kernel basis, when a linear map is injective, surjective or an isomorphism, why the inverse of a bijective linear map is linear, and the rank of a composition.
Video 5
Sums, Direct Sums and Projections
Sums and Intersections of Subspaces
How to add and intersect subspaces, why their sum usually differs from their union, the dimension formula for a sum proved from a basis of the intersection, and the product of two vector spaces.
Video 6
Direct Sums and Projections
Direct sums and why their vectors split uniquely, how a complementary subspace defines a projection, why idempotent maps are exactly these projections, and how oblique projections differ from orthogonal ones.
Video 7