given system, find solns, show solns form vector space, and

[testing]

I need to find the set of solutions to a system of linear equations, show that they are a vector space of R^4, and find a basis for them. The equations are:

x₁ - 2x₃ - 3x₄ = 0
x₂ - 3x₃ + x₄ = 0
2x₁ - x₂ - x₃ - 7x₄ = 0

I got a solution of x = x₃*(2, 3, 1, 0). I could show closure, but how do you find a basis with just the one vector?

[Martingale]

I need to find the set of solutions to a system of linear equations, show that they are a vector space of R^4, and find a basis for them. The equations are:

x₁ - 2x₃ - 3x₄ = 0
x₂ - 3x₃ + x₄ = 0
2x₁ - x₂ - x₃ - 7x₄ = 0

I got a solution of x = x₃*(2, 3, 1, 0). I could show closure, but how do you find a basis with just the one vector?

you should have two vectors that form your basis

(2,3,1,0) is correct

and the other is (3,-1,0,1)