## Question

The functions f_1(x)=3x-1, f_2(x)=4x and f_3(x)=2x are linearly dependent. Show this by finding values of c_1, c_2 and c_3, not all zeros, such that c_1f_1(x)+c_2f_2(x)+c_3f_3(x)=0.

f_1(x)=3x-1, f_2(x)=4x, f_3(x)=2x.

The task is to find such constants c_1, c_2, c_3 not equal zero simultaneously such that

c_1f_1(x)+c_2f_2(x)+c_3f_3(x)=0.

c_1(3x-1)+c_2*4x+c_3*2x=0

x(3c_1+4c_2+2c_3)-c_1=0.

From this following system is obtained:

{(3c_1+4c_2+2c_3=0),(c_1=0):}.

From second equation c_1=0 and first equation gives 4c_2+2c_3=0 or c_3=-2c_2.

So, if c_1=0, c_2=t, then c_3=-2t (t in R) and c_1f_1(x)+c_2f_2(x)+c_3f_3(x)=0.

In particular, we can take c_1=0, c_2=1, c_3=-2.