Question

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decision.

Answer 1

The question is incomplete. The complete question is :

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decision.

$\{ x^5, x^5-1,3 \} \text{ on } (- \infty, \infty)$

Solution :

Given :

Function : $\{ x^5, x^5-1,3 \} \text{ on } (- \infty, \infty)$

We have to determine whether the given function is linear dependent or linearly independent for the interval $(-\infty, \infty)$.

The given function are linearly dependent because for the constants, $c_1$ and $c_2$, the equation is :

$c_1x^5 + c_23 = x^5-1$  has the solution $c_1 = 1$ and $c_2 = -\frac{1}{3}$

Therefore,

$1x^5 + \left(-\frac{1}{3}\right)3 = x^5-1$