All bases considered in these are assumed to be ordered bases.

Below are various vectors in cartesian, cylindrical and spherical coordinates. Express the given vectors in two other coordinate systems outside the coordinate system in which they are expressed
$a)\overrightarrow{A}(x,y,z)={\overrightarrow{e}}_x$
$d)\overrightarrow{A}(\rho ,\varphi ,z)={\overrightarrow{e}}_{\rho }$
$g)\overrightarrow{A}(r,\theta ,\varphi )={\overrightarrow{e}}_{\theta }$
$j)\overrightarrow{A}(x,y,z)=\frac{-y{\overrightarrow{e}}_x+x{\overrightarrow{e}}_y}{x^2+y^2}$

[ k 1 −2 ,4 −1 2]

Consider the elliptical-cylindrical coordinate system (eta, psi, z), defined by $x=a\cos h\eta \cos \psi ,y=a\sin h\eta \sin \psi ;z=z,\eta GE0,0LE\psi LE2\pi ,zR.InPS6$
it was shown that this is an orthogonal coordinate system with scale factors ${\displaystyle h_1=h_2=a{({\text{cosh}}^2\eta -{\cos }^2\psi )}^{\frac{1}{2}}.}$
Determine the dual bases ${\displaystyle (E1,E2,E3),(\eta ,\eta \psi ,z).Showt\widehat{:}f=a\frac{1}{a}\frac{{({\text{cosh}}^{2}eat-{\cos }^s\psi )}^1}{2}[\frac{f}{\eta }e1+\frac{f}{\psi }e2+\frac{f}{z}e3,\frac{f}{w}here(e1,e2,e3)}$ denotes the unit coordinate basis.

Given the elow bases for $R^2$ and the point at the specified coordinate in the standard basis as below, (40 points)

$(B1=\{(1,0),(0,1)\}$&

$B2=(1,2),(2,-1)\}$(1, 7) = $3^{\ast }(1,2)-(2,1)$
$B2=(1,1),(-1,1)(3,7=5^{\ast }(1,1)+2^{\ast }(-1,1)$
$B2=(1,2),(2,1)(0,3)=2^{\ast }(1,2)-2^{\ast }(2,1)$
$(8,10)=4^{\ast }(1,2)+2^{\ast }(2,1)$
B2 = (1, 2), (-2, 1) (0, 5) =
(1, 7) =

a. Use graph technique to find the coordinate in the second basis. (10 points) b. Show that each basis is orthogonal. (5 points) c. Determine if each basis is normal. (5 points) d. Find the transition matrix from the standard basis to the alternate basis. (15 points)