Null space and column space

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

Introduction to the null space of a matrix

Null space 2: Calculating the null space of a matrix

Null space 3: Relation to linear independence

Column space of a matrix

Null space and column space basis

Visualizing a column space as a plane in R3

Proof: Any subspace basis has same number of elements

Dimension of the null space or nullity

Dimension of the column space or rank

Showing relation between basis cols and pivot cols

Showing that the candidate basis does span C(A)