How Do You Find a Basis for the Null Space of Matrix A?

[Derill03]

Matrix A:

1 2 4 1
2 4 8 2
3 1 5 7

The question says find a basis for the solution set AX=0, X is the vector of variables
[x1,x2,x3,x4]^t

What is a basis? and how can i approach this problem?

 

[Mark44]

By inspection, I can see that the solution space for the equation AX = 0 will be at least a one-dimensional subspace of R⁴ (i.e., a line through the origin), and on closer inspection I can see that this solution space will be a two-dimensional subspace of R⁴, a plane through the origin.

You asked what a basis is. Isn't that term defined in your textbook? What this problem is asking for is a set of vectors that spans the solution space. IOW, a set of vectors such that any solution vector is a linear combination of the basis vectors.

You should also look up the definitions of the terms I have underlined.

 

[HallsofIvy]

Solve the equations x+ 2y+ 4z+ u= 0, 2x+ 4y+ 8z+ u= 0, and 3x+ y+ 5z+ 7u= 0. There is, of course, an infinite number of solutions so instead of a single solution you will get equations expressing some of the variables in terms of the others. Choose simple values for those "others" and solve for the rest.