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Need help understanding how to get to this linear algebra answer

Find the dimension and a basis of the following vector spaces V over the given field K.
(i) V is the set of all vectors (α,β,γ) in R^3 with α + 2β −2γ = 0; K = R.
(ii) V is the set of all vectors (α,β,γ) in R^3 with α − β = −γ and α + 2β = γ; K = R.


I have that (i) = Dimension of two, basis is (2,1,0), (-4,0,1) and (ii) is of dimension one with a basis being (1,-2,3)

Anyone have any idea of the steps they used to get these answers?
Original post by pineapplechemist
Find the dimension and a basis of the following vector spaces V over the given field K.
(i) V is the set of all vectors (α,β,γ) in R^3 with α + 2β −2γ = 0; K = R.
(ii) V is the set of all vectors (α,β,γ) in R^3 with α − β = −γ and α + 2β = γ; K = R.


I have that (i) = Dimension of two, basis is (2,1,0), (-4,0,1) and (ii) is of dimension one with a basis being (1,-2,3)

Anyone have any idea of the steps they used to get these answers?

First one: the equation is that of a plane. That's manifestly two-dimensional. All you need is two linearly independent vectors which span that plane. Can you produce such vectors?

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