Linear algebra, Math 2R3
Bradd Hart
Sept. 8, 2011
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Bradd Hart
Review of Complex Numbers
Short-term outline
Review of complex numbers, sections 10.1 - 10.3
Review of real vector spaces, sections 5.1 - 5.4
Introduce complex vector spaces, section 10.4
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Bradd Hart
Review of Complex Numbers
The complex numbers
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Bradd Hart
Review of Complex Numbers
The complex numbers
Introduce a new quantity, i, such that i 2 = −1.
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Bradd Hart
Review of Complex Numbers
The complex numbers
Introduce a new quantity, i, such that i 2 = −1.
The complex numbers are then all expressions of the form
a + bi where a and b are real numbers.
logo
Bradd Hart
Review of Complex Numbers
The complex numbers
Introduce a new quantity, i, such that i 2 = −1.
The complex numbers are then all expressions of the form
a + bi where a and b are real numbers.
Operations on the complex numbers
logo
Bradd Hart
Review of Complex Numbers
The complex numbers
Introduce a new quantity, i, such that i 2 = −1.
The complex numbers are then all expressions of the form
a + bi where a and b are real numbers.
Operations on the complex numbers
Addition:
(a + bi) + (c + di) = (a + c) + (b + d)i
logo
Bradd Hart
Review of Complex Numbers
The complex numbers
Introduce a new quantity, i, such that i 2 = −1.
The complex numbers are then all expressions of the form
a + bi where a and b are real numbers.
Operations on the complex numbers
Addition:
(a + bi) + (c + di) = (a + c) + (b + d)i
Multiplication:
(a + bi) · (c + di) = (ac − bd) + (ad + bc)i
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Bradd Hart
Review of Complex Numbers
Multiplicative inverse
Every non-zero complex number has a multiplicative inverse.
That is, if z1 is not zero then the equation, in the unknown z,
z1 z = 1 has a solution.
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Bradd Hart
Review of Complex Numbers
Multiplicative inverse
Every non-zero complex number has a multiplicative inverse.
That is, if z1 is not zero then the equation, in the unknown z,
z1 z = 1 has a solution.
The conjugate
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Bradd Hart
Review of Complex Numbers
Multiplicative inverse
Every non-zero complex number has a multiplicative inverse.
That is, if z1 is not zero then the equation, in the unknown z,
z1 z = 1 has a solution.
The conjugate
If z = a + bi then z̄, the conjugate of z, is a − bi.
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Bradd Hart
Review of Complex Numbers
Multiplicative inverse
Every non-zero complex number has a multiplicative inverse.
That is, if z1 is not zero then the equation, in the unknown z,
z1 z = 1 has a solution.
The conjugate
If z = a + bi then z̄, the conjugate of z, is a − bi.
Notice that z z̄ = a2 + b2 so
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Bradd Hart
Review of Complex Numbers
Multiplicative inverse
Every non-zero complex number has a multiplicative inverse.
That is, if z1 is not zero then the equation, in the unknown z,
z1 z = 1 has a solution.
The conjugate
If z = a + bi then z̄, the conjugate of z, is a − bi.
Notice that z z̄ = a2 + b2 so
z̄
1
=
z
z z̄
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Bradd Hart
Review of Complex Numbers
The complex plane
y
6
r(a, b)
r
θ
- x
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Bradd Hart
Review of Complex Numbers
The complex plane
y
6
r(a, b)
r
θ
- x
√
r = a2 + b2 ; this is called the modulus of the complex
number z = a + bi and written |z|.
logo
Bradd Hart
Review of Complex Numbers
The complex plane
y
6
r(a, b)
r
θ
- x
√
r = a2 + b2 ; this is called the modulus of the complex
number z = a + bi and written |z|.
We saw that z · z̄ = |z|2 .
logo
Bradd Hart
Review of Complex Numbers
The complex plane
y
6
r(a, b)
r
θ
- x
√
r = a2 + b2 ; this is called the modulus of the complex
number z = a + bi and written |z|.
We saw that z · z̄ = |z|2 .
θ is an argument for a + bi and is only determined up to
multiples of 2π.
logo
Bradd Hart
Review of Complex Numbers
The complex plane
y
6
r(a, b)
r
θ
- x
√
r = a2 + b2 ; this is called the modulus of the complex
number z = a + bi and written |z|.
We saw that z · z̄ = |z|2 .
θ is an argument for a + bi and is only determined up to
multiples of 2π.
a = r cos(θ) and b = r sin(θ) so z = r (cos(θ) + i sin(θ)).
Bradd Hart
Review of Complex Numbers
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