Unit 3 of 11

Vector Spaces

  • 5 lessons
  • Videos 13 to 17

A subspace is a set of vectors that contains the zero vector and is closed under addition and scaling. This unit tests subspaces and independence, then finds bases, dimension and rank.

Spans and the solution sets of homogeneous systems are the first examples, while a shifted plane and a union of two lines show what can fail. Linear independence makes coefficients unique, a basis is an independent list that spans, and dimension counts the vectors in any basis. The unit ends with the four fundamental subspaces of a matrix, the column space, row space, null space and left null space, with bases and dimensions found by elimination.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Subspaces, Independence and Bases

  1. Subspaces, Sums and Intersections

    When a set of vectors is a subspace, how the subspace test checks zero and closure under addition and scaling, and why spans, homogeneous solution sets, sums and intersections pass while affine sets and unions can fail.

    Video 13

  2. Linear Independence

    What makes a list of vectors linearly independent or dependent, how a dependence relation shows which vector is redundant, why independence makes coefficients unique, and how to test a list by elimination.

    Video 14

  3. Bases and Dimension

    What a basis is, how an ordered basis gives each vector a coordinate vector, how to prune a spanning list or extend an independent list to a basis, and why dimension, the size of any basis, is well defined.

    Video 15

The Four Fundamental Subspaces

  1. Column Space, Row Space and Rank

    How to find bases for the column space and the row space of a matrix by elimination, why basis columns come from the original matrix, why row rank equals column rank, and what a rank factorization is.

    Video 16

  2. Null Space and Left Null Space

    How to find bases for the null space and the left null space of a matrix, how the rank and the size of the matrix fix the nullity and the left nullity, and where each of the four fundamental subspaces lives.

    Video 17