Determine whether the given set of functions is linearly ind

Determine whether the given set of functions is linearly independent on the interval (,).

(a) f1(x) = sin x, f2(x) = 0, f3(x) = ln x

(b) f1(x) = 3, f2(x) = sin^2 (x), f3(x) = cos^2 (x)

(c) f1(x) = 1 + x, f2(x) = 2x, f3(x) = x^2

Answers:

(a) linearly dependent ((0)f1 + (1)f2 + (0)f3 = 0)

(b) linearly dependent ((1/3)f1 + (1)f2 + (-1)f3 = 0)

(c) linearly independent

Solution

a)

Linearly dependent because

0*sin(x)+c*0+0*ln x=0

for any non zero real number c

b)

We have the identity:

sin^2(x)+cos^2(x)=1

Hence,

3+3*(-sin^2(x))-3 cos^2(x)=3-3(sin^2(x)+cos^2(x))=3-3=0

Hence linearly dependent

c)

Let, a,b,c so that

af1+bf2+cf3=0

a(1+x)+2bx+cx^2=0

Hence, c=0

a=0

a+2b=0 hence, b=0

Hence set is linearly independent.