Solutions of Homogeneous System

[zohapmkoftid]

Homework Statement

If X₀ and X₁ are solutions to the homogeneous system of equation AX = 0, show that rX₀ + sX₁ is also a solution for any scalars r and s.

Thanks for help!

Homework Equations

The Attempt at a Solution

 

[Mark44]

Homework Statement

If X₀ and X₁ are solutions to the homogeneous system of equation AX = 0, show that rX₀ + sX₁ is also a solution for any scalars r and s.

From the given information, what can you say about the values of Ax₀ and Ax₁?

 

[zohapmkoftid]

Since X₀ and X₁ are the solutions, therefore
AX₀ = 0 and AX₁ = 0
Right?

 

[Mark44]

Right. Now what is A(rx₀ + sx₁)?

You need to know something about how matrix multiplication works.

 

[zohapmkoftid]

A(rX₀ + sX₁)
= ArX₀ + AsX₁
= rAX₀ + sAX₁
= 0

Thanks for your help

 

[zohapmkoftid]

But how can we prove rX₀ + sX₁ = 0 from A(rX₀ + sX₁) = 0

 

[Inferior89]

But how can we prove rX₀ + sX₁ = 0 from A(rX₀ + sX₁) = 0

This is not what you are supposed to prove.

You are supposed to prove that X = rX₀ + sX₁ is a solution to the equation AX = 0. You just did that.

 

[zohapmkoftid]

Thanks. I understand now

 

[Susanne217]

Homework Statement

If X₀ and X₁ are solutions to the homogeneous system of equation AX = 0, show that rX₀ + sX₁ is also a solution for any scalars r and s.

Thanks for help!

Homework Equations

The Attempt at a Solution

You guys forgot to mention that this fact is the socalled super-position principle for ODEs.

Which states if $$x_1, x_2$$ are solution of the ODE $$x' = F(t,x)$$

which states that a solution x can be formed of two existing solutions

such that $$x(t) = r \cdot x_1(t) + s \cdot x_2(t)$$ is also a solution of the ODE.

This is also called a linear combination for you young wipping snappers out there.

You can read about it here http://tutorial.math.lamar.edu/Classes/DE/SecondOrderConcepts.aspx

 

[HallsofIvy]

I have no idea why you would think anyone "forgot" to say that or why it would need saying. What is given here is a basic property of linear transformations and is used in many applications other that linear differential equations. (There is NO "super-position" principle for general ODEs.)