Question

show the functions f(t) = cos t, g(t) =In t, and h(t) are linearly independent by considering linear combinations

Answer 1

What is the function h(t)?

Answer 2

opps h(t) = t squared

Answer 3

oops*

Answer 4

To show that those 3 functions are linearly independent, you need to show that:
\[c_1\cos(t)+c_2\ln(t)+c_3t^{2} = 0 \Rightarrow c_1,c_2,c_3 = 0\]
In other words, if you have some linear combination of those functions that equals 0, then all the constants must be 0, you only have the trivial solution.

Answer 5

To show that 3 functions f, g, and h are linearly independent, we have to show that the linear combination\[C_1f(x) + C_2g(x) + C_3h(x) = 0\] only when \[C_1, C_2, C_3 = 0\]

We have \[C_1cost + C_2lnt + C_3t^2 = 0\]

It should be obvious that there is no way to combine these terms to get 0 with non-zero coefficients.

Answer 6

That statement should should be true for any values of t, so lets plug in some values to see if we can prove all the constants are 0.
Gonna right it out on paper and post it, one sec

Answer 7

@Joemath, unfortunately that's not true. The linear combination can have roots with non-zero coefficients; linear independence means that there is no non-zero set of coefficients for which the function is 0 at all t.

Answer 8

(I wish it were; it would make proving linear independence MUCH easier.)

Answer 9

I believe what joemath314159 is saying is that if \[C_1, C_2, C_3\]are constants such as\[C_1\cos{t}+C_2\log{t}+C_3t^2 = 0\]then this must be satisfied for any t, so, for example,\[C_1 \cos{1} + C_3 = 0\]\[C_1 + C_2\log{(\pi/2)}+C_3\pi^2/4 = 0\]\[C_2\log{\pi}+C_3\pi^2 = 0\]
This linear system has only one solution:\[C_1 = C_2 = C_3 = 0\]so the functions are linearly independent. This is correct because it shows the only set of constants that makes the lineal combination zero for all t is a set of zeros.

Answer 10

Oh, yes, I guess I misunderstood what he wrote.

Answer 11

Yes, using the property that this must be true for all 0 and showing numerical evidence otherwise is the way to solve it.

Answer 12

Krebante's solution is much more beautiful than mine lol

Answer 13

I mistyped some equations though. The second one should be\[C_2 \log{(\pi/2)} + C_3 \pi^2/4 = 0\]and the third one\[-C_1+C_2\log{\pi}+C_3\pi^2 = 0.\]

Answer 14

We had the same idea though, basically you want to plug in 3 different values of t (try to be clever with your choices as to knock out a couple of functions) and show that the only solution to the resulting system leads to all coefficients being 0.

Answer 15

Thank-you all!!