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Nullspace (Kernel matrix) without nullspace function

Null Space (also Kernel matrix) is the set of all vectors v of matrix M for which Mv=0.

How to construct nullspace matrix in maxima?

Given matrix:

Matrix

in maxima:

m:matrix([1,1,-1],[1,5,1],[1,-1,-2]);

In Calculator

The null space of this matrix consists of all vectors (x, y, z) ∈ R3 for which:

The null space of this matrix consists of all vectors

This can be written as a system of linear equations:

System of linear equations

in maxima:

eq1:x+y-z=0; eq2:x+5*y+z=0; eq3:x-y-2*z=0;

In Calculator

We can solve these equations

Solve equations

in maxima:

linsolve([eq1,eq2,eq3],[x,y,z]);

In Calculator

We can write the null space (solution to Mv = 0) in terms of c, where c is scalar

Nullspace

Since c is a free variable this can be simplified for example to

simplified Nullspace

Verification in maxima:

Verification

n:matrix([3/2],[-(1/2)],[1]);

m.n

This Example in Calculator

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Nullspace

Matrix

Linsolve