Chapter 3, Problem 10CR

(a)

To determine

Whether the set of function $f_1\left(x\right)=\ln x,f_2\left(x\right)=\ln x^2$ is linearly independent or not on the interval $\left(0,\infty \right)$.

(b)

To determine

Whether the set of function $f_1\left(x\right)=x^n,f_2\left(x\right)=x^{n+1},n\in \mathbb{N}$ is linearly independent or not on the interval $\left(-\infty ,\infty \right)$.

(c)

To determine

Whether the set of function $f_1\left(x\right)=x,f_2\left(x\right)=x+1$ is linearly independent or not on the interval $\left(-\infty ,\infty \right)$.

(d)

To determine

Whether the set of function $f_1\left(x\right)=\cos \left(x+\frac{\pi }{2}\right),f_2\left(x\right)=\sin x$ is linearly independent or not on the interval $\left(-\infty ,\infty \right)$.

(e)

To determine

Whether the set of function $f_1\left(x\right)=0,f_2\left(x\right)=x$ is linearly independent or not on the interval $\left(-5,5\right)$.

(f)

To determine

Whether the set of function $f_1\left(x\right)=2,f_2\left(x\right)=2x$ is linearly independent or not on the interval $\left(-\infty ,\infty \right)$.

(g)

To determine

Whether the set of function $f_1\left(x\right)=x^2,f_2\left(x\right)=1-x^2,f_3\left(x\right)=2+x^2$ is linearly independent or not on the interval $\left(-\infty ,\infty \right)$.

(h)

To determine

Whether the set of function $f_1\left(x\right)=x{e}^{x+1},f_2\left(x\right)=\left(4x-5\right)e^x,f_3\left(x\right)=x{e}^x$ is linearly independent or not on the interval $\left(-\infty ,\infty \right)$.