Basis for the nullspace of this matrix

[Noxide]

this is apparently "really simple", but I just don't know how to do it from the examples I have and I feel like a moron...

what's the basis for the nullspace of this matrix

[ 2 3 1]
[ 5 2 1]
[ 1 7 2]
[ 6 -2 0]

 

[phyzmatix]

I'm not sure if your notation is a list of vectors in the matrix or if that is the actual matrix so I can't be more exact for now than to remind you that the nullspace is the solution space of the system Ax = 0, so start off by solving for the coefficient matrix A augmented with 0. Remember also that the basis of a space is the set of linearly independent vectors that spans the space in question.

 

[HallsofIvy]

this is apparently "really simple", but I just don't know how to do it from the examples I have and I feel like a moron...

what's the basis for the nullspace of this matrix

[ 2 3 1]
[ 5 2 1]
[ 1 7 2]
[ 6 -2 0]

The nullspace is the set of vectors, <x, y, z> such that
$$\begin{bmatrix}2 & 3 & 1 \\ 5 & 2 & 1\\ 1 & 7 & 2\\6 & -2 & 0\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}0 \\ 0 \\ 0 \end{bmatrix}$$
That is the same as the four equations 2x+ 3y+ z= 0, 5x+ 2y+ z= 0, x+ 7y+ 2z= 0, and 6x- 2y= 0. If we subtract the first equation from the second, we get 3x- y= 0 or y= 3x. Putting that into 6x- 2y= 0 we get 6x- 2(3x)= 6x- 6x= 0 which is satisfied for all x.

If we subtract 2 times the first equation from the third we get -3x+ y= 0 or y= 3x again. Those equations all reduce to y= 3x which cannot be reduced further. Putting y= 3x into the first equation we get 2x+ 3(3x)+ z= 0 so z= -11x. Putting y= 3x into the second equation we get 5x+ 2(3x)+ z= 0 so z= -11x. Putting y= 3x into the third equation, we get x+ 7(3x)+ 2z= 0 so 2z= -22x and z= -11x.

That is, any solution of that equation, any vector in the kernel, can be written as <x, y, z>= <x, 3x, -11x>= x <1, 3, -11>.