Finding a basis for a particular subspace with Dot Product restrictions

[alingy1]

Find the basis of the subspace of R4 that consists of all vectors perpendicular to both [1, -2, 0, 3] and [0,2,1,3].

My teacher applies dot product: Let [w,x,y,z] be the vectors in the subspace. Then,

w-2x+3z=0 and 2x+y+3z=0

So, she solves the system and get the following:

Subspace= { t[-1,-1/2,1,0] + s[-6,-3/2,0,1]|t,s are in R}.

But, I do the following:

I isolate w and y: w=2x-3z and y=-2x-3z.

I replace them : Supspace= { [2x-3z,x,-2x-3z,z]|x,z are in R} = span{[2,1,-2,0],[-3,0,-3,1]}.

I set up a system of linear equation to see if [-3,0,-3,1] is a linear combination of the vectors in my teacher's answer. However, it is not.

What am I doing wrong?

 

[RUber]

It is a linear combination of the teacher's answer.
You did not do anything wrong with your solution.
Let t=-3, s=1.