Understanding Null Space: A Plain English Guide

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Discussion Overview

The discussion revolves around the concept of null space in linear algebra, specifically seeking a plain English explanation. Participants explore its definition, geometric interpretation, and implications in various contexts, including ordinary matrices and fields.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant requests a plain English explanation of null space.
  • Another participant defines null space as the set of vectors x such that Ax = 0, noting it is a subspace of the domain.
  • A geometric interpretation is provided, stating that the null space consists of vectors perpendicular to the row space of A.
  • One participant suggests that the null space represents all solutions to a homogeneous system of equations.
  • A counterpoint is raised, arguing that not all vector spaces have the concept of perpendicularity, using a specific matrix example to illustrate this point.
  • Another participant expresses confusion about the field F_2 and clarifies their focus on ordinary matrices with numerical entries.
  • A clarification is made that for Ax = 0 to hold true, x must be perpendicular to each row of A in the context of real numbers.
  • F_2 is identified as the field with two elements, relevant to the earlier discussion.
  • A participant agrees that the earlier statement holds true for vectors and matrices with real entries.

Areas of Agreement / Disagreement

Participants express differing views on the geometric interpretation of null space, particularly regarding the concept of perpendicularity in various contexts. The discussion remains unresolved on this point.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about vector spaces and the definitions of perpendicularity, particularly in different fields.

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could someone kinda explain in plain english what null space is?
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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.
 
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.
 
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.
 
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
 
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]
 
Last edited:
F_2 is the field with two elements, though any field of characteristic two would do.
 
All right... anyway the statement is true for vectors and matrices with entries in R with the usual definitions of everything.
 
Last edited:
wow~
thank you very much!
 

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