# What is the use of Null Space?

This answer explains what is null space very effectively. In brief Null Space is the set of vectors which have 0 effect on the system when applied. So, what is the use of finding null-space? Is it just that it gives us what not to use and whether the matrix is invertible or not or is there a better use for null space? May be something like, "we know adding null-space-vectors won't change the system but improves the stability of the system?" (I'm just guessing)

Any practical examples (like the ones given in the answer referred) are greatly appreciated. Theoretical ones will also be helpful.
Thanks

• The existence of an 'identity element' is typically a crucial axiom for many mathematical structures (in the context of matrices, the zero matrix is the additive identity) . Without an identity element, you could not have a well defined group, or vector space, and so on. The existence of an identity element is also useful in proofs. – Bill Wallis Apr 29 '18 at 8:45

## 2 Answers

Dear fellow mathematician,

at first, it might appear that it is not really as useful. But the opposite is true, you are going to use it to find eigenvectors which are of huge importance in linear algebra. They are used for diagonalising matrices and for Singular Value Decomposition which is of vital importance. (it’s a numerically stable way of taking powers of matrices - solving differential equations and many other things - i.e. finding n-th term of a sequence from a reccurent formula, which can be used for various)

These are the applications that I’m familiar with, I’m sure that someone will provide more.

Have a nice Sunday!

As Bill Wallis said, the existence of a zero is important, which is provided by the vectors in the null space. In all practical applications, it is very important that the solutions don't have a component in the null space, which has no effect on the system. Having a component in the null space only wastes resources, which we don't want to do. This is also evident from the answer quoted.
Example 1: Fuel is wasted if thrusters are fired in the null space.
Example 2: Investment leads to no gain of profit and hence wasted in the null space.
Example 3: Power applied in the null space does not illuminate the room further, and hence power is wasted.

Due to this nulling effect, it is also used in simplification techniques, like the ones mentioned by innerz09. I am sure that there are other reasons as well.

Cheers!!!

• So, by knowing what is there in null space, we get to know what not to use. That's it right? Is there no practical use of null-space vectors? – Nagabhushan S N Apr 30 '18 at 8:19
• Simplification is one such use. There may be other uses as well. – Aaditya Ravindran Apr 30 '18 at 8:53