# Finding a basis for vector spaces

I'm having trouble finding a basis for algebraically defined vector spaces where there is more than one condition. For instance, I can easily find a basis for the vector space in R^3 defined by a+2b+3c=0 (where a,b,c are the elements of the vector), but I have no idea what to do when the vector space is defined by something like a-b-c=0 & 2a+3b+8c=0.

For the first example I would write that every vector in the space has to be of the form (-2b-3c, b, c) = b(-2,1,0) + c(-3,0,1), so a basis for the vector space would be {(-2,1,0),(-3,0,1)}. I have no idea what to do for the second example.

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