Answer
See below
Work Step by Step
Given $\{1,x,x^2,x^3\}$
a) Write a linear combination of these functions
$\lambda_1(1)+\lambda_2(x)+\lambda_3(x^2)+\lambda_4(x^3)=0\\
\lambda_1+\lambda_2 x+\lambda_3 x^2+\lambda_4 x^3=0+0(x)+0(x^2)+0(x^3)$
which is equal to $\lambda_1=\lambda_2=\lambda_3=\lambda_4=0$
b) For $(1,x,...,x^k)$ we have:
$\lambda_1(x)+\lambda_2(x)+...+\lambda_{k+1}(x^k)=0\\
\rightarrow \lambda_1=\lambda_2=...=\lambda_{k+1}=0$
Hence $\{(1,x,...,x^k)\}$ is linearly independent.