Question

Determine if the pair of functions is L.I or L.D.
f (t) = t ^ 2 + 5t, g (t) = t ^ 2 - 5t

Answer 1

Are there any constants \(c_1,c_2\) for which the following holds?
\[c_1f(t)+c_2g(t)=0\]

Answer 2

If yes, then L.I. If no, then L.D.

Answer 3

so you are asking if you can make g(t) from a linear combination of f(t)?

Answer 4

I need to use the Wronskian?

Answer 5

\[W(f(t),g(t))=\begin{vmatrix}f(t)&g(t)\\f'(t)&g'(t)\end{vmatrix}=f(t)g'(t)-f'(t)g(t)\]
If the Wronskian \(\not=\) 0, then f and g are linearly independent.

Answer 6

But the same conclusion should be obtained using my other method...
\[c_1(t^2+5t)+c_2(t^2-5t)=0\\
c_1t^2+5c_1t+c_2t^2-5c_2t=0\\
(c_1+c_2)t^2+5(c_1-c_2)t=0\]
You get the system of equations
\[\begin{cases}
c_1+c_2=0\\
c_1-c_2=0
\end{cases}\]

Answer 7

I'm thinking they are linearly independent: there is no a s.t. a*f(t) = g(t) and vice versa.

Answer 8

Yep, the system above yields \(c_1=c_2=0\), so the functions are linearly independent.

Likewise, the Wronskian yields \(10t^2\not=0\) for all \(t\in\mathbb{R}\), so lin. ind.

Answer 9

Thanks.

Answer 10

You're welcome