Math 1433W Elementary Linear Algebra 19 October 2020 Warm-up poll: - - PowerPoint PPT Presentation
Math 1433W Elementary Linear Algebra 19 October 2020 Warm-up poll: Expand and Instructor: dr Adam Abrams simplify http://theAdamAbrams.com/teaching (2 + i )(2 i ) . Warm-up poll (2 + i )(2 i ) Expand and simplify . (2+i)(2-i) =
19 October 2020 Instructor: dr Adam Abrams http://theAdamAbrams.com/teaching
Warm-up poll: Expand and simplify .
(2 + i)(2 − i)
Expand and simplify .
(2 + i)(2 − i)
(2+i)(2-i) = (2)(2) + (2)(-i) + (i)(2) + (i)(-i) = 4 - 2i + 2i + 1 = 5
No problem session today. Moved to next week, 26 October, at 11:15 am. After that, 16 November and all Even Mondays from then on. Extra lecture. Please fill out the Doodle link from yesterday’s email.
Re Im
labels for the “real axis” and ”imaginary axis"
Multiply the two complex numbers shown below.
Re Im
Which of the colored points is ?
z⋅w
A B C D E w z
Re Im
Which of the colored points is ?
z⋅w
A B C D E w z
Re Im
Rectangular form Polar form Picture Formula
a
b
Im Re
z r
θ
Re Im
z
Re(z)
Im(z)
Im Re
z |z|
arg(z)
Re Im
z
Rectangular form Polar form
The modulus of a complex number is written and is the distance from to the point . The argument of a complex number is written
and is the counter-clockwise (CCW) angle from the axis to the point . We usually require . If , then and .
z |z| 0 = 0 + 0i z z arg(z) arg z +Re z −180∘ < arg(z) ≤ 180∘ z = r cis(θ) |z| = r arg(z) = θ
By default, 0° points to the right, and angles are measured counter-clockwise from there.
0∘ 30∘ 45∘ 60∘ 90∘ 120∘ 135∘ 150∘ 180∘ 210∘ 225∘ 240∘ 270∘ 300∘ 315∘ 330∘ 360∘ −30∘ −45∘ −60∘ −90∘
120∘ 45∘
If you want to use radians, you can. It’s not required.
π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π 7π/6 5π/4 4π/3 3π/2 5π/3 7π/4 11π/6 2π −π/6 −π/4 −π/3 −π/2
2π/3 π/4
30° 60°
3
“45-45-90 triangle” “30-60-90 triangle”
45° 45° 1
30°
60°
3 2
45° 45°
2 2 2 2
What is in rectangular form?
cis(210∘)
Re Im
What is in rectangular form?
cis(210∘)
3 2 1 2
Re Im
What is in rectangular form?
cis(210∘)
3 2 1 2
Answer: −
3 2 + −1 2 i − 3 2 − 1 2i − 3 2 − i 1 2 − 3 − i 2
Re Im
any of these are acceptable formats
What is in rectangular form?
2 cis(45∘)
Re Im
What is in rectangular form?
2 cis(45∘)
45∘ 2 45∘
Re Im
Answer:
What is in rectangular form?
2 cis(45∘)
2 2
Re Im
2
Multiplying by a complex number by a positive real number stretches (or scales) the modulus of by a factor of .
z s z s s(r cis θ) = (sr)cis θ
Re Im
×2
Re Im
−2 − 2i −4 − 4i −1 + 2i −2 + 4i 1+ 3
2 i
2 + 3i
2 3 −i 4 3 −2i
We know that multiplying a number by the number gives a number that is rotated counter-clockwise from . In general, multiplying by rotates counter-clockwise by . We know that multiplying by real scales lengths. Therefore, multiplying by scales by and rotates CCW by . As a formula:
z i = cis(90∘) 90∘ z z cis(θ) z θ r > 0 r cis(θ) r θ
We know that multiplying a number by the number gives a number that is rotated counter-clockwise from . In general, multiplying by rotates counter-clockwise by . We know that multiplying by real scales lengths. Therefore, multiplying by scales by and rotates CCW by . As a formula:
z i = cis(90∘) 90∘ z z cis(θ) z θ r > 0 r cis(θ) r θ
|z + w| arg(z + w) Re(z ⋅ w) Im(z ⋅ w) are ugly
Which of the colored points is ?
z⋅w
A B C D E w z
Re Im
Which of the colored points is ?
z⋅w
A B C D E w
Re Im
z
Because is a real number ( ), the number has the same argument as itself.
With two complex numbers, we can add to get the sum subtract to get the difference multiply to get the product divide to get the quotient With one complex number we can negate to get the negative reciprocate to get the reciprocal
conjugate to get the conjugate z
The complex conjugate of a complex number is the reflection of across the real axis. It is written and spoken as “z bar”.
z z z
Re Im
The complex conjugate of a complex number is the reflection of across the real axis. It is written and spoken as “z bar”.
z z z
Re Im
z
z
w
w
u
u
r
r = r
The complex conjugate of a complex number is the reflection of across the real axis. What are polar and rectangular formulas for conjugates?
z z
Re Im
right
a
up
b
z = a + bi z
down
up”
b −b
z = a − bi
z r
The complex conjugate of a complex number is the reflection of across the real axis. What are polar and rectangular formulas for conjugates?
z z
ccw
θ
Re Im
z = r cis(θ)
cw or ccw
θ −θ
z = r cis(−θ)
and and
z + w = z + w z − w = z − w zw = z w ( z w ) = z w
if w≠0
z = z z z = |z|2
Why? Using rectangular form, (a+bi)(a-bi) = a2 - bi + bi + (bi)(-bi) = a2 + b2 This is |z|2 by Pythagorean Theorem.
and and
z + w = z + w z − w = z − w zw = z w ( z w ) = z w
if w≠0
z = z z z = |z|2
Why? Using polar form, (rcisø)(rcis(-ø)) = r2cis(ø-ø) = r2cis(0) = r2
We only used the rectangular explanation during lecture. Both are valid, but polar is even easier! r2 is literally |z|2.
Write in rectangular form ( ).
3 2 + 5i a + bi
3 2 + 5i = 3 2 + 5i ⋅ 1 = 3 2 + 5i ⋅ 2 − 5i 2 − 5i = 3(2 − 5i) 22 + 52 = 6 − 15i 29 = 6 29 + −15 29 i
This is the same as multiplying by 1. It does not change the value of the original number.
Given that
, what are formulas for ?
z = a + bi z = r cis θ z−1 = 1 z (a + bi)−1 = a − bi a2 + b2 (r cis θ)−1 = ⋯
First, let’ s talk about positive powers in polar form.
What is
?
= ___ ___
What about
?
z2 z15 z100
z1/2 = z z1/3 =
3 z
(rcisØ)2 = (rcisØ)(rcisØ) = (rr)cis(Ø+Ø) = r2cis(2Ø)
Remember:
(r cis θ)(s cis ϕ) = (rs)cis(θ+ϕ)
What is
?
= ___ ___
What about
?
z2 z15 z100
z1/2 = z z1/3 =
3 z
(rcisØ)3 = (rcisØ)2(rcisØ) = (r2cis(2Ø))(rcisØ) = (r2r)cis(2Ø+Ø) = r3cis(3Ø)
Remember:
(r cis θ)(s cis ϕ) = (rs)cis(θ+ϕ)
What is
?
= ___ ___
What about
?
z2 z15 z100
z1/2 = z z1/3 =
3 z
(rcisØ)2 = r2cis(2Ø) (rcisØ)3 = r3cis(3Ø)
Remember:
(r cis θ)(s cis ϕ) = (rs)cis(θ+ϕ)
What is
?
= ___ ___
What about
?
z2 z15 z100
z1/2 = z z1/3 =
3 z
Remember:
(r cis θ)(s cis ϕ) = (rs)cis(θ+ϕ)
For any integer , .
What is
?
= ___ ___
What about
?
z2 z15 z100
z1/2 = z z1/3 =
3 z
Remember:
(r cis θ)(s cis ϕ) = (rs)cis(θ+ϕ)
What is
?
= ___ ___
What about
?
z2 z15 z100
z1/2 = z z1/3 =
3 z
Remember:
(r cis θ)(s cis ϕ) = (rs)cis(θ+ϕ)
Re Im
z = 5 cis(60∘) We want a number w
for which w2 = 5cis(60°). r2cis(2Ø) = 5cis(60°) r = 5 and Ø = 30° r = - 5 and Ø = 30°
5cis(30∘) − 5cis(30∘) = 5cis(−120∘)
What is
?
= ___ ___
What about
?
z2 z15 z100
z1/2 = z z1/3 =
3 z
Remember:
(r cis θ)(s cis ϕ) = (rs)cis(θ+ϕ)
Re Im
z = 5 cis(60∘) = 5 cis(420∘) 5cis(30∘) − 5cis(30∘) = 5cis(−120∘)
Can we use r = 5 and Ø = 210°? Yes, 5cis(210°) = 5cis(-120°).
= 5cis(210∘)
For a positive real number , we require to be positive. For a negative real number , we say that . For a complex number with we say that .
x x −x −x = i x z = r cis θ −180∘ < θ ≤ 180∘ z = ( r)cis( θ
2 )
For a positive real number , we require to be positive. Although , only is the value of . For a negative real number , we say that . Although , only is the value of . For a complex number with we say that . Although , only is .
x x (−3)2 = 9 3 9 −x −x = i x (−4i)2 = −16 4i −16 z = r cis θ −180∘ < θ ≤ 180∘ z = ( r)cis( θ
2 )
( 5 cis(210∘))
2 = 5 cis 60∘
5 cis 30∘ 5 cis 60∘
For a positive real number , we require to be positive. For a negative real number , we say that . For a complex number with we say that .
x
n x
−x
n−x = i n x
z = r cis θ −180∘ < θ ≤ 180∘
n z = ( n r)cis( θ
n )
For positive real numbers, , but this is not guaranteed to be true for complex numbers.
xy = x ⋅ y
Write in rectangular form.
2 − 3i + 1 − 2i i + 2
2−3i + 1 − 2i i + 2 = 2−3i + 1 − 2i i + 2 ⋅ i − 2 i − 2 = 2−3i + (1 − 2i)(i − 2) (i + 2)(i − 2) = 2−3i + i − 2 + (−2i)i + (−2i)(−2) i2 + 2i − 2i − 4 = 2−3i + i − 2 + 2 + 4i −1 − 4 = 2−3i + 5i −5 = 2−3i + (−i) = 2 − 4i
Solve for real and .
(1 + i)x + (1 − 2i)y = 1 − i x y
+ +
x xi y − 2yi = 1−i
We need the real parts of left/right sides to equal equch other and the imaginary parts of left/right sides to equal each other.
Solve for real and .
(1 + i)x + (1 − 2i)y = 1 − i x y
We need the real parts of left/right sides to equal equch other and the imaginary parts of left/right sides to equal each other.
x + y = 1
and
x − 2y = − 1 + +
x xi y − 2yi = 1−i
Solve for real and .
(1 + i)x + (1 − 2i)y = 1 − i x y
We need the real parts of left/right sides to equal equch other and the imaginary parts of left/right sides to equal each other.
x + y = 1
and
x − 2y = −1 y = 1 − x + +
x xi y − 2yi = 1−i
x − 2(1 − x) = −1 x − 2 + 2x = −1 3x = 1 x = 1/3 y = 1 − 1/3 y = 2/3