Understanding Null Space: A Plain English Guide

[asdf1]

could someone kinda explain in plain english what null space is?
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[HallsofIvy]

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.

 

[0rthodontist]

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.

 

[JasonRox]

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.

 

[matt grime]

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

\left( \begin{array}{cc} 1&1\\ 1&1\end{array}\right)

the vector

\left(\begin{array}{c} 1&1 \end{array}\right)

is, after transposition, in the row space of the matrix yet it is also in the kernel if we consider this over F_2

 

[0rthodontist]

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
\left( \begin{array}{cc} 1&1\\ 1&1\end{array}\right) \left(\begin{array}{c} 1&1 \end{array}\right)

is
\left(\begin{array}{c} 2&2 \end{array}\right)

 

[matt grime]

F_2 is the field with two elements, though any field of characteristic two would do.

 

[0rthodontist]

All right... anyway the statement is true for vectors and matrices with entries in R with the usual definitions of everything.

 

[asdf1]

wow~
thank you very much!