Understanding Null Space: A Plain English Guide

In summary: in summary, null space is the set of all vectors perpendicular to each vector in the row space of a matrix.
  • #1
asdf1
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could someone kinda explain in plain english what null space is?
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  • #2
The "null space" of a linear operator, A, also called the "kernel" of the operator, is the set of all vectors. x, in the domain of the operator such that Ax= 0. One can show that that set is a subspace of the domain.
 
  • #3
Also, geometrically, if you're dealing with matrices, then the null space of A is the set of all vectors perpendicular to each vector in the row space of A.
 
  • #4
The easiest way to do it is like this...

Assuming you know how to solve a matrix of a homogeneous system.

If you remember solving it, you will probably remember the parameters it has. Therefore, if it has paramaters, you have infinite many solutions. You can one, but never none.

The nullspace is the set of all solutions to the homogeneous system you just solved.
 
  • #5
0rthodontist said:
Also, geometrically, if you're dealing with matrices, then the null space of A is the set of all vectors perpendicular to each vector in the row space of A.

no it isn't. not all vector spaces even have the concept of perpendicular. ignoring any problems with row vectors v column vectors, consider

[tex]\left( \begin{array}{cc} 1&1\\ 1&1\end{array}\right)[/tex]

the vector

[tex]\left(\begin{array}{c} 1&1 \end{array}\right)[/tex]

is, after transposition, in the row space of the matrix yet it is also in the kernel if we consider this over F_2
 
  • #6
I don't know what F_2 means... I was talking about ordinary matrices with entries that are numbers. In that case for Ax = 0 to be true, then x dot each row of A must be 0, so x is perpendicular to Row A. The only interpretation I know for
[tex]\left( \begin{array}{cc} 1&1\\ 1&1\end{array}\right) \left(\begin{array}{c} 1&1 \end{array}\right)[/tex]

is
[tex]\left(\begin{array}{c} 2&2 \end{array}\right)[/tex]
 
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  • #7
F_2 is the field with two elements, though any field of characteristic two would do.
 
  • #8
All right... anyway the statement is true for vectors and matrices with entries in R with the usual definitions of everything.
 
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  • #9
wow~
thank you very much!
 

1. What is null space?

Null space, also known as the kernel, is the set of all vectors that are mapped to the zero vector by a given linear transformation. In simpler terms, it is the set of all solutions or inputs that result in an output of zero.

2. Why is understanding null space important?

Understanding null space is important because it helps us to understand the behavior of a linear transformation. It also allows us to determine the dimension of the vector space, and provides insight into the types of solutions a system of linear equations may have.

3. How is null space related to the concept of linear independence?

Null space and linear independence are closely related because the null space contains all of the linearly dependent vectors in a vector space. In other words, a vector is in the null space if and only if it can be written as a linear combination of other vectors in the space.

4. Can null space be empty?

Yes, it is possible for null space to be empty. This occurs when a linear transformation maps all vectors in the vector space to a non-zero vector, meaning there are no inputs that result in an output of zero. In this case, the null space is said to be trivial.

5. How do you find the null space of a matrix?

The null space of a matrix can be found by solving the homogeneous system of linear equations represented by the matrix. This can be done using techniques such as Gaussian elimination or finding the reduced row echelon form. The solutions to this system will give the basis for the null space.

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