WEEK 
Monday 
Wednesday 
Friday 
week1  822 
824 
826 
week 2  829 
831 
92 
week 3  95 
97 
99 
week 4 
912 
911 
913 
week 5 
919 
921 
923 
week 6 
926 
928 
930 
week 7 
103 
105 
107 
week 8 
1010 
1012 
1014 
week 9  1017 
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1021 
week 10 
1024 
1024 
1026 
week 11 
1031 
112 
114 
week 12 
117 
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1111 
week 13 
1114 
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week
14 NoClasses 
1121 
1123 
1127 
week 15 
1128 
1130 
122 
week 16 
125 
127 
119 
week 17 Final Exam 
1212 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 e^{x} = 1 + x +
x^{2}/2! +x^{3}/3! + ...
then e^{ix} = 1 + ix + (ix)^{2}/2!
+(ix)^{3}/3! + ... = 1 + ix  x^{2}/2!
 ix^{3}/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 codomain, 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?