Linear independence of functions: $x_1(t) = 3$, $x_2(t)=3\sin^2t$, $x_3(t)=4\cos^2t$

Solution 1:

Yes, indeed, your answer is fine. And it would have been a particularly fine determining the linear (in)dependence of a system of equations that doesn't readily admit of another observation about the relationship between $\cos^2 t$ and $\sin^2 t$ $(\dagger)$. Indeed, you're one step away from working with the Wronskian, which is a useful tool to prove linear independence.

$(\dagger)$ Now, to the observation previously noted: You could have also used the fact that $$x_1(t) - \left[(x_2(t) +\frac 34 x_3(t)\right] = 3 - (3 \sin^2 t + 3\cos^2 t)= 3 - 3\left(\underbrace{\sin^2(t) + \cos^2(t)}_{\large = 1}\right)=0$$

and saved yourself a little bit of work: you can read off the nonzero coefficients $c_i$ to demonstrate their existence: $c_1 = 1, c_2 = -1, c_3 = -\frac 34$, or you could simply express $x_1$ as a linear combination of $x_2, x_3$, to conclude the linear dependence of the vectors. (But don't count on just any random set of vectors turning out so nicely!)

Solution 2:

It is much more easier to use known identity $\sin^2{t}+\cos^2{t}=1$. We have $x_1(t)-x_2(t)-\frac{3}{4}x_3(t)=3-3\cos^2{t}-3\sin^2{t}=0$, so functions are linearly dependent.

Solution 3:

This is a pretty straightforward question, firstly, let me remind you the definition of linearly dependent functions. It says,

A set of functions $\mathrm{f_1(x), f_2(x), ... , f_n(x)}$ are called linearly dependent if

$\mathrm{c_1f_1(x)+ c_2f_2(x)+ ... + c_n f_n(x) = 0, where \ c_1, c_2, ... , c_n \ are \ arbitrary \ constants}$ holds true for atleast two non-zero c's.

Coming to the question, then

$\mathrm{f_1(x) = 3, \ f_2(x) = 3sin^2x, \ f_3(x) = 4cos^2x}$

Consider $\mathrm{c_1, c_2, c_3}$ as arbitrary constants,

$\mathrm{3c_1 + 3c_2sin^2x + 4c_3cos^2x}$

Easily, if $\mathrm{c_1 = \frac{-1}{3}, \ c_2 = \frac{1}{3}, \ c_3 = \frac{1}{4}}$, then

$\mathrm{3c_1 + 3c_2sin^2x + 4c_3cos^2x}$ = $0$

As all the arbitrary constants are non-zero, the functions are definitely LD.