Null space : Linear Independence

In this blog, i'm going to talk about a null space when column vectors in a matrix are linearly independent.

 

1. Linear Independence : Before we get into the topic, I want to review the definition of linear independence quickly. The linear independence is vectors in a set that cannot be represented by any linear combination of the other vectors. The null space is a set of vectors that satisfies this equation.

So, The null space is also a subspace, which means that it contains the zero vector and it's closure under any scalar multiplication and vector addition.

 

2. Linera Independence and a null space : Then, we want to know what is a solution for the equation when column vectors in a matrix are linearly independent.

Here's what i have in this example. Matrix a is a m by n matrix. Instead of writing it a matrix, I just turned the matrix into an array of column vectors and I assumed that the column vectors are linearly independent.

Here, I multiplied matrix A with vector x, which has m components, and it results in the zero vector. It represents that I want to know about the null space of this situation. I said that the column vectors are linearly independent. And when they are linearly independent, it won't be able to represent one vector as combining other vectors. So, in order to meet the equation, the only solution to solve the question is that all the components in vector x have to be zero.