Question

For each of the following, show that the given vectors are linearly independent in C[-1,1]:

(a)cosπx, sinπx
(b)x³,|x|³

Answer 1

To demonstrate the vectors cosπx and sinπx, as well as x³ and |x|³ are linearly independent, one must show that the only solution to the linear combinations equalling zero for all x in [-1,1] is when all coefficients are zero.

Step-by-step explanation:

To show that vectors are linearly independent, we must prove that no linear combination of these vectors equals the zero vector unless all coefficients in the combination are zero.

For example, for the vectors cosπx and sinπx, we would need to show that if a*cosπx + b*sinπx = 0 for all x in [-1,1], then a and b must both be zero.

For the vectors x³ and |x|³, we must show a similar statement: if a*x³ + b*|x|³ = 0 for all x in [-1,1], then a and b must both be zero. Checking this involves examining the behavior of the functions for negative, zero, and positive values of x and showing that the only solution is a=b=0.