Question

Determine whether the given functions are linearly dependent or linearly independent on the interval$\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{)}$ .

(a)$\left\{e^{2x},x^2{e}^{2x},e^{-x}\right\}$

(b)$\left\{e^{x}sin2x,x{e}^{x}sin2x,e^x,x{e}^x\right\}$

(c) $\left\{2e^{2x}-e^x,e^{2x}+1,e^{2x}-3,e^x+1\right\}$

Short answer

1. Given set of functions are Linearly Independent (LI).
2. Given set of functions are Linearly Independent (LI).
3. Given set of functions are Linearly Independent (LI).

Step-by-step solution

Determine the given functions are linearly dependent or linearly independent

$<e^{2x},x^2{e}^{2x},e^{-x}>$

Assuming scalars$c_1,c_2,c_3$such that

$c_1{e}^{2x}+c_2{x}^2{e}^{2x}+c_3{e}^{-x}=0$

At

$$\begin{gathered} x=0 \\ {c}_4=0=c_2 \\ {c}_1+c_3=0 \end{gathered}$$

At$x\to \infty$

$$\begin{gathered} c_1(\infty )+c_2(\infty )=0 \\ {c}_1=c_2=0=c_3 \end{gathered}$$

Hence

Given set of functions are Linearly Independent (LI).

Determine the given functions are linearly dependent or linearly independent

$<e^x\sin 2x,x{e}^x\sin 2x,e^x,x{e}^x>$

Assuming scalars$c_1,c_2,c_3,c_4$such that

$c_1{e}^x\sin 2x+c_{2}x{e}^x\sin 2x+c_3{e}^x+c_{4}x{e}^x=0$

At$x\to 0$

$c_4=0=c_2$

At x = 0

$$\begin{gathered} c_3=0 \\ {c}_1=c_2=c_3=c_4=0 \end{gathered}$$

Hence

Given set of functions are Linearly Independent (LI).

Determine the given functions are linearly dependent or linearly independent

$<2e^{2x}-e^x,e^{2x}+1,e^{2x}-3,e^x+1>$

Assuming scalars $c_1,c_2,c_3,c_4$such that

$c_1\left(2e^{2x}-e^x\right)+c_2\left(e^{2x}+1\right)+c_3\left(e^{2x}-3\right)+c_4\left(e^x+1\right)=0$

At $x\to -\infty$

$c_2-2c_3+c_4=0$

At x = 0

$c_1+2c_2-2c_3+2c_4=0$

At x = 1

$e^2\left(2c_1+c_2+c_3\right)=c_1-c_1$

At x = -1

$\left.2c_1+c_2+c_3+c_{\left(c_1\right.}-c_1\right)=0$

From equation third and fourth we get

$$\begin{gathered} c_2=-c_1 \\ {c}_4=c_1 \\ {c}_3=-c_1 \end{gathered}$$

From equation second, we can get

$$\begin{gathered} c_1=0 \\ {c}_1=c_2=c_3=c_4=0 \end{gathered}$$

Hence

Given set of functions are Linearly Independent (LI).