Complex Numbers and 
Inner Products : 
Fourier Transform warm-up

Jim Mahoney, Marlboro College, Feb 2011

1.	7 (e^(0.3 i))= x + i y .  Find x,y .

2.	7 + 0.3 i = R (e^(i θ)). Find R, θ .

3.	 (2 + i) (3 - i) = ?

4.	 Convert the complex numbers in the last problem to polar form.  Show that in the multiplication, the angles add.

5.	(((2 + 3 i))^((1/3))) = ?

6.	 cos(1 + i) = ?

7.	 (i^i) = ?

Define three vectors as follows :
		(Subscript[(OverHat[e]), 1])= ((1/2), (1/2), (1/(Sqrt[2]))) 
		(Subscript[(OverHat[e]), 2])= ((1/2), (1/2), -(1/(Sqrt[2]))) 
		(Subscript[(OverHat[e]), 3])= ((1/(Sqrt[2])), -(1/(Sqrt[2])), 0)

8.	Show that these three vectors form an ortho-normal basis.  In other words, that each has unit length, 
	and that they’re all perpendicular.  (Hint: all of this is done with the inner “dot” product.)  
	Sketch a picture of where they look like in the traditional (x,y,z) space.

9.	Let 
		(Overscript[a, →])  = (OverHat[i]) + 2 (OverHat[j]) + 3 (OverHat[k])  = (1, 2, 3)
	Find the coordinates of  (Overscript[a, →]) in the (OverHat[e]) basis.  
	That is, find three numbers (Subscript[a, 1]), (Subscript[a, 2]), (Subscript[a, 3]) such that
		(Overscript[a, →])  = (Subscript[a, 1])(Subscript[(OverHat[e]), 1]) + (Subscript[a, 2])(Subscript[(OverHat[e]), 2]) + (Subscript[a, 3])(Subscript[(OverHat[e]), 3]) = Σ (Subscript[a, i])(Subscript[(OverHat[e]), i])

Are we having fun yet?