# 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

### 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

### 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).