Proving Solutions of Linear Systems: A Plane in R^n

[hkus10]

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]

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. And follow the hint.

 

[hkus10]

Yes. And follow the hint.

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

Is this correct?

 

[LCKurtz]

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.

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

 

[LCKurtz]

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...

 

[hkus10]

Now try something else along those lines...

What is the goal for that?

 

[LCKurtz]

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.