# Vector Spaces

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?

LCKurtz
Homework Helper
Gold Member
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?

Ap + A(su) + A(tv) = b
Ap + s(Au) + t(Av) = b
Ap + s(0) + t(0) = b
Ap = b

Is this correct?

Last edited:
LCKurtz
Homework Helper
Gold Member
what should I go from here?

Sorry. No more help from here until you show us what you have tried following the hint. Show us your effort.

 Your post hadn't shown up when I wrote this. See my next post.

Last edited:
LCKurtz
Homework Helper
Gold Member
Ap + A(su) + A(tv) = b
Ap + s(Au) + t(Av) = b
Ap + s(0) + t(0) = b
Ap = b

Is this correct?

Yes. You have now shown that p is a solution to the NH equation. Now try something else along those lines...

Now try something else along those lines...
What is the goal for that?

LCKurtz
Homework Helper
Gold Member
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?

Yes. You have now shown that p is a solution to the NH equation. Now try something else along those lines...

What is the goal for that?

Because you aren't done. See the red above.