True/False Question on Linear Combos

[NullSpace0]

Homework Statement

If the system Ax=b has two known solutions, x₁ and x₂, true or false that another solution exists. If it does, find another solution.

Homework Equations

The Attempt at a Solution

I think it's true. But after that, I don't know.

I'm really lost, but I think I remember learning that given two solutions, any linear combination of the two is also a solution. Does that mean that x₁+x₂ is a solution?

I'm worried that that is actually a solution to Ax=2b, though.

 

[NullSpace0]

Perhaps that is only true for homogeneous equations, now that I think about it.

Would nonhomogeneous equations require some sort of weighted average linear combination?

 

[SammyS]

Homework Statement

If the system Ax=b has two known solutions, x₁ and x₂, true or false that another solution exists. If it does, find another solution.

Homework Equations

The Attempt at a Solution

I think it's true. But after that, I don't know.

I'm really lost, but I think I remember learning that given two solutions, any linear combination of the two is also a solution. Does that mean that x₁+x₂ is a solution?

I'm worried that that is actually a solution to Ax=2b, though.

Is it true that (1/2)Ax = A((1/2)x) ?

 

[LCKurtz]

Homework Statement

If the system Ax=b has two known solutions, x₁ and x₂, true or false that another solution exists. If it does, find another solution.

.

Is it given b is not the zero vector? Think about uniqueness, rank, and free variables.