Suppose that the solution set to a linear system Ax = b is a plane

in R^n with vector equation x = p + su + tv , s, t ∈ R . Prove that

p is a solution to the nonhomogeneous system Ax = b , and that

u and v are both solutions to the homogeneous system Ax = 0 .

(Hint Try choices of s and t).

 

Should I start from A(p + su + tv) = b? If yes, what should I do from here? If no, where should I start?

The first part is more trivial than you probably expect it. For the 2nd part of the question (Au=0 and Av=0), your start is indeed promising. Keep in mind what "linear" means for that one.

Ap + A(su) + A(tv) = b

Ap + s(Au) + t(Av) = b

Ap + s(0) + t(0) = b

Ap = b

 

Is this correct?

Ap + A(su) + A(tv) = b

Ap + s(Au) + t(Av) = b

Ap + s(0) + t(0) = b

Ap = b

 

Is this correct?

 

You shoulde be able to determine for yourself when you have a valid proof for a given proposition.

 

But,, no, this is not a valid proof for the proposition that you stated. Your statements in fact assume the conclusion of the proposition.