WEEK |
Monday |
Wednesday |
Friday |
week1 | 8-22 |
8-24 |
8-26 |
week 2 | 8-29 |
8-31 |
9-2 |
week 3 | 9-5 |
9-7 |
9-9 |
week 4 |
9-12 |
9-11 |
9-13 |
week 5 |
9-19 |
9-21 |
9-23 |
week 6 |
9-26 |
9-28 |
9-30 |
week 7 |
10-3 |
10-5 |
10-7 |
week 8 |
10-10 |
10-12 |
10-14 |
week 9 | 10-17 |
10-19 |
10-21 |
week 10 |
10-24 |
10-24 |
10-26 |
week 11 |
10-31 |
11-2 |
11-4 |
week 12 |
11-7 |
11-9 |
11-11 |
week 13 |
11-14 |
11-16 |
11-18 |
week
14 NoClasses |
11-21 |
11-23 |
11-27 |
week 15 |
11-28 |
11-30 |
12-2 |
week 16 |
12-5 |
12-7 |
11-9 |
week 17 Final Exam |
12-12 Exam |
`A^n =A` if `n` is odd. |
|
|
`A^n` `->` |
( |
|
) |
A= | ( |
|
) |
The geometry of complex arithmetic:
If z = a+bi = ||z||(cos(t) +i sin(t)) and w = c+di = ||w||(cos(s) +i sin(s)) then
z+w = (a+c)+(b+d)i which corresponds geometrically to the "vector " sum of z and w in the plane, and
zw = ||z||(cos(t) +i sin(t)) ||w||(cos(s) +i sin(s))=
||z|| ||w|| (cos(t) +i sin(t))(cos(s) +i
sin(s))
= ||z|| ||w|| (cos(t) cos(s) -
sin(t)sin(s) + (sin(t) cos(s) + sin(s)cos(t)) i)
= ||z|| ||w|| (cos(t+s) + sin(t+s) i)
So you use the product of the magnitudes of z and w to determine the magnitude of the product and use the sum of the angles to determine the angle of the product.
Notation: cos(t) + i sin(t)
is somtimes written as cis(t).
Note: If we consider the series for ex = 1 + x +
x2/2! +x3/3! + ...
then eix = 1 + ix + (ix)2/2!
+(ix)3/3! + ... = 1 + ix - x2/2!
- ix3/3! + ...
... = cos(x) + i sin(x)
Thus `e^{i*p} = cos(pi) + i sin(pi)= -1`. So `ln(-1) =
i *p`.
Furthermore: `e^{a+bi} = e^a*e^{bi} = e^a ( cos (b) + sin(b) i)
`
Matrices with complex number entries.
If r and s are complex numbers in the matrix A, then as n
get large if ||r|| < 1 and ||s|| < 1 the powers of A
will get close to the zero matrix , if r=s=1 the
powers of A will always be A, and otherwise the powers
of A will diverge .
Polynomials with complex coefficients.
Because multiplication and addition make sense for complex
numbers, we can consider polynomials with coefficients that
are complex numbers and use a complex number for the
variable, making a complex polynomial a function from the
complex numbers to the complex numbers.
This can be visualized using one plane for the domain of
the polynomial and a second plane for the co-domain, target,
or range of the polynomial.
The Fundamental Theorem of Algebra: If f
is a non constant polynomial with complex number
coefficients then there is at
least on complex number z* where f(z*)
= 0.
For more on complex numbers see: Dave's Short Course on Complex Numbers,
|
|
How are these questions related to Motivation Question I?