- #1

- 57

- 8

**x**'=A

**x**

Let A be 2x2 for simplicity. Then the general solution would look like:

**x**(t) = α

**a**+ β

**b**

And a fundamental matrix would be:

Ψ(t) = (

**a**,

**b**)

What confuses me is this: I tried making a new fundamental matrix by replacing the first column of Ψ(t) by a linear combination of the general solution, something like:

**x**(t) = 2

**a**+ 4

**b**

Now my new fundamental matrix looked like this:

Ψ(t) = ( 2

**a**+ 4

**b**,

**b**)

And expanding the following expression:

**x**(t)=Ψ(t)

**c**, where

**c**is the vector of constants, I found out that I get the same general solution

**x**(t), with different eigenvectors (however they were simply scalar multiples of the eigenvectors of the matrix A)

My question is this, are linear combinations of the fundamental set of solutions also a fundamental set of solutions? Like, would

{ 2

**a**+ 4

**b**,

**b**}

also be a fundamental set of solutions? I guess it would because they are linearly independent... If not, why do we call Ψ(t) a fundamental matrix when we can build one using linear combinations of the fundamental set of solutions? All these questions confuse me, I just need some clarification.

Thanks in advance!