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

What else do you need to prove?In summary, 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. From the given hint, we can 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 by using choices of s and t. This is shown by the steps Ap + A(su) + A(tv) = b, Ap + s(Au) + t(Av) = b, Ap + s(0) + t(0) = b, and Ap = b. However, further proof is needed to
  • #1
hkus10
50
0
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?
 
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  • #2
hkus10 said:
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.
 
  • #3
LCKurtz said:
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?
 
Last edited:
  • #4
hkus10 said:
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.
 
Last edited:
  • #5
hkus10 said:
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...
 
  • #6
LCKurtz said:
Now try something else along those lines...
What is the goal for that?
 
  • #7
hkus10 said:
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 said:
Yes. You have now shown that p is a solution to the NH equation. Now try something else along those lines...

hkus10 said:
What is the goal for that?

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

1. What is a linear system?

A linear system is a set of equations that can be represented in the form of Ax = b, where A is a matrix of coefficients, x is a vector of variables, and b is a vector of constants. The goal is to find the values of x that satisfy all of the equations in the system.

2. What does it mean to prove a solution of a linear system?

To prove a solution of a linear system means to show that the values of x that are found to satisfy the equations in the system are indeed the correct values. This can be done by substituting the values into the equations and verifying that they hold true.

3. What is a plane in R^n?

A plane in R^n is a two-dimensional flat surface in n-dimensional space. It can be represented by a set of equations, and a linear system can be used to find the intersection of multiple planes in R^n.

4. How can a linear system be solved?

A linear system can be solved using various methods such as elimination, substitution, or matrix operations. The goal is to reduce the system to a simpler form where the values of x can be easily determined.

5. Why is proving solutions of linear systems important?

Proving solutions of linear systems is important because it ensures that the values of x found are correct and satisfy all of the equations in the system. This is necessary for making accurate predictions and solving real-world problems in various fields such as physics, engineering, and economics.

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