Null space and column space

12 videos
We will define matrix-vector multiplication and think about the set of vectors that satisfy Ax=0 for a given matrix A (this is the null space of A). We then proceed to think about the linear combinations of the columns of a matrix (column space). Both of these ideas help us think the possible solutions to the Matrix-vector equation Ax=b.

Matrix vector products

VIDEO 21:10 minutes
Defining and understanding what it means to take the product of a matrix and a vector

Introduction to the null space of a matrix

VIDEO 10:23 minutes
Showing that the Null Space of a Matrix is a valid Subspace

Null space 2: Calculating the null space of a matrix

VIDEO 13:07 minutes
Calculating the null space of a matrix

Null space 3: Relation to linear independence

VIDEO 11:35 minutes
Understanding how the null space of a matrix relates to the linear independence of its column vectors

Column space of a matrix

VIDEO 10:40 minutes
Introduction to the column space of a matrix

Null space and column space basis

VIDEO 25:13 minutes
Figuring out the null space and a basis of a column space for a matrix

Visualizing a column space as a plane in R3

VIDEO 21:11 minutes
Determining the planar equation for a column space in R3

Proof: Any subspace basis has same number of elements

VIDEO 21:35 minutes
Proof: Any subspace basis has same number of elements

Dimension of the null space or nullity

VIDEO 13:59 minutes
Dimension of the Null Space or Nullity

Dimension of the column space or rank

VIDEO 12:48 minutes
Dimension of the Column Space or Rank

Showing relation between basis cols and pivot cols

VIDEO 8:33 minutes
Showing that linear independence of pivot columns implies linear independence of the corresponding columns in the original equation

Showing that the candidate basis does span C(A)

VIDEO 13:40 minutes
Showing that just the columns of A associated with the pivot columns of rref(A) do indeed span C(A).