Determine whether the given pair of functions is linearly independent or linearly dependent.?

Question

f(t) = t g(t) = 1/t

So.... i think i might have the answer. So, if someone could tell me if i'm right or wrong, that would be great.

(I'll use a and b as constants here)

at + b/t = 0

a = 1 and b = -t

So this would be linearly independant since a and b do not have to equal zero.

What i am concerned about is if i can use t as a constant or not.

Thanks!

Answer 1

two functions are linearly dependent only if they are constant multiples of each other......but f(t) means t is a variable..not a constant

Answer 2

Your approach is wrong because the constants you pick must work for all t. You cannot use t as a constant. Here is how you do it.

We want to find out for which a and b it is true that af(t)+bg(t)=0 for ALL t. Since this equation must hold for all t, it must hold for 2 values of t in particular. Here, the choice of values is arbitrary, so you can just pick any 2 convenient values. I'm picking t=1 and t=2. Let t=1. We must have a+b=0. Let t=2. We must have 2a+b/2=0. If you solve the simultaneous equations a+b=0 and 2a+b/2=0, you get a=b=0. Hence, f and g are linearly independent.