6.1.3-4 Show that the given columns of f(x) are linearly independent on the interva;

Nov 2009
717
133
Wahiawa, Hawaii
$\textsf{Show that the given columns of functions are linearly independent on the interval
$(-\infty,\infty)$}$
3.
$$\left[\begin{array}{c}e^{2x}\cos{3x} \\ -e^{2x}\sin{3x} \end{array}\right]
,\left[\begin{array}{c}e^{2x}\sin{3x} \\ -e^{2x}\cos{3x} \end{array}\right]$$
ok clearly new at this but thot to start with
$$\left[\begin{array}{c}e^{2x}\cos{3x} \\ -e^{2x}\sin{3x} \end{array}\right]
+\left[\begin{array}{c}e^{2x}\sin{3x} \\ -e^{2x}\cos{3x} \end{array}\right]
=\left[\begin{array}{c}0 \\ 0 \end{array}\right]$$
------------------------------------------------------------------------------
4.
$$\left[\begin{array}{c}e^{-2x}\\0\\0 \end{array}\right],
\left[\begin{array}{c}0\\ 3\cos{5x}\\ -3\sin{3x} \end{array}\right]
,\left[\begin{array}{c} 0\\ \sin{3x}\\ \cos{5x} \end{array}\right]$$
 

Walagaster

MHF Helper
Apr 2018
223
144
Tempe, AZ
Re: 6.1.3-4 Show that the given columns of f(x) are linearly independent on the inter

$\textsf{Show that the given columns of functions are linearly independent on the interval
$(-\infty,\infty)$}$
3.
$$\left[\begin{array}{c}e^{2x}\cos{3x} \\ -e^{2x}\sin{3x} \end{array}\right]
,\left[\begin{array}{c}e^{2x}\sin{3x} \\ -e^{2x}\cos{3x} \end{array}\right]$$
ok clearly new at this but thot to start with
$$\left[\begin{array}{c}e^{2x}\cos{3x} \\ -e^{2x}\sin{3x} \end{array}\right]
+\left[\begin{array}{c}e^{2x}\sin{3x} \\ -e^{2x}\cos{3x} \end{array}\right]
=\left[\begin{array}{c}0 \\ 0 \end{array}\right]$$
------------------------------------------------------------------------------
4.
$$\left[\begin{array}{c}e^{-2x}\\0\\0 \end{array}\right],
\left[\begin{array}{c}0\\ 3\cos{5x}\\ -3\sin{3x} \end{array}\right]
,\left[\begin{array}{c} 0\\ \sin{3x}\\ \cos{5x} \end{array}\right]$$
Hint: What happens if you try $x=0$ in those?
 
Nov 2009
717
133
Wahiawa, Hawaii
Re: 6.1.3-4 Show that the given columns of f(x) are linearly independent on the inter

so for these if x=0 then
$$\left[\begin{array}{c}1\\0\\0 \end{array}\right],
\left[\begin{array}{c}0\\ 3\\ 0 \end{array}\right]
,\left[\begin{array}{c} 0\\ 0\\ 1 \end{array}\right]$$

would this rref into
$$\left[\begin{array}{c}1&0&0\\0&1&0\\0&0&1 \end{array}\right]$$