Can someone understand this passage of Linear Algebra?

[EngPF]

Homework Statement

Hello there everybody! I'm reading a Linear Algebra textbook, specifically on LTV systems solutions.
I'm trying to redo this example from the book:

Homework Equations

But I couldn't understand the passage:

The Attempt at a Solution

I mean. x1(0) = 1 and x2(0) = 0? I don't know from where that came from. And then x(t) = [1 0.5t^2]^T ?

I can see from the x1(t) = x1(0) and x2(t) = (1/2)(t^2)x1(0) + x2(0) that

x(t) = [1 0.5t^2]^T [x1(0) x2(0)] <= This I can relate!

But the second one where x(t) = [ 1 2 ]^T I can't understand.

I think it might have been arbitrarily choosen values? Either that or I'm missing some basic concept here. I've tried to get back some pages/chapters but could find the answer for that. Should I study the first chapters all again? (yeah probably I should! =/ )

Anyway, could someone help me to understand this?

Thank you very much!

 

[Dick]

Homework Statement

Hello there everybody! I'm reading a Linear Algebra textbook, specifically on LTV systems solutions.
I'm trying to redo this example from the book:

View attachment 75422

Homework Equations

But I couldn't understand the passage:

View attachment 75423

The Attempt at a Solution

I mean. x1(0) = 1 and x2(0) = 0? I don't know from where that came from. And then x(t) = [1 0.5t^2]^T ?

I can see from the x1(t) = x1(0) and x2(t) = (1/2)(t^2)x1(0) + x2(0) that

x(t) = [1 0.5t^2]^T [x1(0) x2(0)] <= This I can relate!

But the second one where x(t) = [ 1 2 ]^T I can't understand.

I think it might have been arbitrarily choosen values? Either that or I'm missing some basic concept here. I've tried to get back some pages/chapters but could find the answer for that. Should I study the first chapters all again? (yeah probably I should! =/ )

Anyway, could someone help me to understand this?

Thank you very much!

I think you are correct that [1,0]^T and [1,2]^T are arbitrarily chosen linearly independent initial conditions. They didn't 'derive' them.