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Q Review Problems-2E

Expert-verifiedFound in: Page 343

Book edition
9th

Author(s)
R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages
616 pages

ISBN
9780321977069

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

- Given set of functions are Linearly Independent (LI).
- Given set of functions are Linearly Independent (LI).
- Given set of functions are Linearly Independent (LI).

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

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

$x=0\phantom{\rule{0ex}{0ex}}{c}_{4}=0={c}_{2}\phantom{\rule{0ex}{0ex}}{c}_{1}+{c}_{3}=0\phantom{\rule{0ex}{0ex}}$

At $x\to \infty $

${c}_{1}(\infty )+{c}_{2}(\infty )=0\phantom{\rule{0ex}{0ex}}{c}_{1}={c}_{2}=0={c}_{3}\phantom{\rule{0ex}{0ex}}$

Hence

Given set of functions are Linearly Independent (LI).

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

Assuming scalars ${c}_{1},{c}_{2},{c}_{3},{c}_{4}$ such that

${c}_{1}{e}^{x}\mathrm{sin}2x+{c}_{2}x{e}^{x}\mathrm{sin}2x+{c}_{3}{e}^{x}+{c}_{4}x{e}^{x}=0$

At $x\to 0$

${c}_{4}=0={c}_{2}$

At x = 0

${c}_{3}=0\phantom{\rule{0ex}{0ex}}{c}_{1}={c}_{2}={c}_{3}={c}_{4}=0$

Hence

Given set of functions are Linearly Independent (LI).

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

Assuming scalars ${c}_{1},{c}_{2},{c}_{3},{c}_{4}$ such that

${c}_{1}\left(2{e}^{2x}-{e}^{x}\right)+{c}_{2}\left({e}^{2x}+1\right)+{c}_{3}\left({e}^{2x}-3\right)+{c}_{4}\left({e}^{x}+1\right)=0$

At $x\to -\infty $

${c}_{2}-2{c}_{3}+{c}_{4}=0$

At x = 0

${c}_{1}+2{c}_{2}-2{c}_{3}+2{c}_{4}=0$

At x = 1

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

At x = -1

$\left.2{c}_{1}+{c}_{2}+{c}_{3}+{c}_{\left({c}_{1}\right.}-{c}_{1}\right)=0$

From equation third and fourth we get

${c}_{2}=-{c}_{1}\phantom{\rule{0ex}{0ex}}{c}_{4}={c}_{1}\phantom{\rule{0ex}{0ex}}{c}_{3}=-{c}_{1}$

From equation second, we can get

${c}_{1}=0\phantom{\rule{0ex}{0ex}}{c}_{1}={c}_{2}={c}_{3}={c}_{4}=0$

Hence

Given set of functions are Linearly Independent (LI).

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