Null space of a matrix introduction

In this chapter, I'm going to talk about null space of a matrix

 

Before we start, Let's review the notions of subspace.

1. What is a subspace ? It's a subset that meets three conditions. 

1. The subset contains zero vector.
2. The subset is closed under scalar multiplication.
3. The subset is closed under vector addition.

If a subset satisfies the conditions, then we say the subset is a subspace.

 

2. Null space : the underlying question for this situation is this. We have a set that satisfies the equation matrix A times vector x is equal to 0. Is this set a subspace?

The first condition is the set contains the zero vector.

It's true. Next, The second condition is that the set is closure under scalar multiplication.

It's also true. The last condition is that the set is closed under vector addition.

As a result, the set N is a valid subspace. We call this set N the null space of A. and we can write it as N(A).