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) = (2)(2) + (2)(-i) + (i)(2) + (i)(-i) = 4 - 2i + 2i + 1 = 5

Scheduling 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.

Calculators Im labels for the “real axis” and ”imaginary axis" Re

Calculators Multiply the two complex numbers shown below. Im Re

Calculators z ⋅ w Which of the colored points is ? Im C D B w A E z Re

Calculators z ⋅ w Which of the colored points is ? Im B C D A E w z Re

Complex numbers Rectangular form Polar form Im Im z z r b Picture θ Re Re a Formula z = a + bi z = r cis( θ )

Complex numbers Rectangular form Polar form Im Im z z | z | Im( z ) arg( z ) Re Re Re( z )

Polar form z | z | The modulus of a complex number is written and is the 0 = 0 + 0 i z distance from to the point . z arg( z ) arg z The argument of a complex number is written or and is +Re z the counter-clockwise ( CCW ) angle from the axis to the point . − 180 ∘ < arg( z ) ≤ 180 ∘ We usually require . | z | = r z = r cis( θ ) arg( z ) = θ If , then and .

Measuring angles By default, 0° points to the right, and angles are 
 measured counter-clockwise from there. 90 ∘ 120 ∘ 60 ∘ 135 ∘ 45 ∘ 120 ∘ 30 ∘ 150 ∘ 45 ∘ 180 ∘ 360 ∘ 0 ∘ 210 ∘ 330 ∘ − 30 ∘ 225 ∘ 315 ∘ − 45 ∘ 240 ∘ 300 ∘ 270 ∘ − 60 ∘ − 90 ∘

Measuring angles If you want to use radians, you can. It’s not required. π /2 2 π /3 π /3 3 π /4 π /4 2 π /3 5 π /6 π /6 π /4 π 0 2 π 7 π /6 11 π /6 − π /6 5 π /4 7 π /4 − π /4 4 π /3 5 π /3 3 π /2 − π /3 − π /2

Right triangles 60° 2 45° 1 2 1 45° 30° 1 3 “45-45-90 triangle” “30-60-90 triangle”

Right triangles 1 1 1 2 60° 45° 2 2 30° 45° 3 2 2 2 Memorize these numbers!

Polar → Rectangular cis(210 ∘ ) What is in rectangular form? Im Re cis(210 ∘ )

Polar → Rectangular cis(210 ∘ ) What is in rectangular form? Im 3 2 3 1 2 Re 1 2 2 1 cis(210 ∘ )

Polar → Rectangular cis(210 ∘ ) What is in rectangular form? Im 3 Answer: − + − 1 2 i 2 3 − 1 3 2 i − 2 any of these Re 2 1 2 are acceptable cis(210 ∘ ) 3 − i 1 − formats 2 2 3 − i − 2

Polar → Rectangular 2 cis(45 ∘ ) What is in rectangular form? Im Re

Polar → Rectangular 2 cis(45 ∘ ) 45 ∘ 2 What is in rectangular form? Im 2 cis(45 ∘ ) 45 ∘ Re

Polar → Rectangular 2 cis(45 ∘ ) What is in rectangular form? Im 2 1 2 cis(45 ∘ ) 2 2 2 2 2 Re 2 2 + i 2 Answer:

Multiplication in polar form z s Multiplying by a complex number by a positive real number z s stretches (or scales) the modulus of by a factor of . s ( r cis θ ) = ( sr )cis θ Im Im − 2 + 4 i 2 + 3 i × 2 − 1 + 2 i 1+ 3 2 i Re Re 2 3 − i 4 3 − 2 i − 2 − 2 i − 4 − 4 i

Multiplication in polar form i = cis(90 ∘ ) z We know that multiplying a number by the number gives a 90 ∘ z number that is rotated counter-clockwise from . z cis( θ ) z θ In general, multiplying by rotates counter-clockwise by . r > 0 We know that multiplying by real scales lengths. r cis( θ ) θ Therefore, multiplying by scales by and rotates CCW by . r As a formula: ( r cis θ ) ⋅ ( s cis ϕ ) = ( r ⋅ s )cis( θ + ϕ )

Multiplication in polar form i = cis(90 ∘ ) z We know that multiplying a number by the number gives a 90 ∘ z number that is rotated counter-clockwise from . z cis( θ ) z θ In general, multiplying by rotates counter-clockwise by . r > 0 We know that multiplying by real scales lengths. r cis( θ ) θ Therefore, multiplying by scales by and rotates CCW by . r As a formula: | z ⋅ w | = | z | ⋅ | w | and arg( z ⋅ w ) = arg( z ) + arg( w )

Re( z ⋅ w ) | z + w | are ugly arg( z + w ) Im( z ⋅ w ) Re( z + w ) = Re z + Re w and Im( z + w ) = Im z + Im w . | z ⋅ w | = | z | ⋅ | w | and arg( z ⋅ w ) = arg( z ) + arg( w )

Calculators z ⋅ w Which of the colored points is ? Im B C D A E w Re z

Calculators z ⋅ w Which of the colored points is ? Because is a real z Im number ( ), arg z = 0 B the number has C wz D the same argument A as itself. w E w Re z

Operations With two complex numbers, we can z + w add to get the sum z − w subtract to get the difference z ⋅ w or zw multiply to get the product z ÷ w or z / w divide to get the quotient With one complex number we can − z negate to get the negative z − 1 or 1/ z reciprocate to get the reciprocal conjugate to get the conjugate z

Complex conjugate z z The complex conjugate of a complex number is the reflection of across the real axis. z It is written and spoken as “z bar”. Im Re

Complex conjugate z z The complex conjugate of a complex number is the reflection of across the real axis. z It is written and spoken as “z bar”. Im z u w r = r r Re w u z

Complex conjugate z z The complex conjugate of a complex number is the reflection of across the real axis. What are polar and rectangular formulas for conjugates? a + bi = a − bi Im z = a + bi b up Re a right b down − b or “ up” z z = a − bi

Complex conjugate z z The complex conjugate of a complex number is the reflection of across the real axis. What are polar and rectangular formulas for conjugates? a − bi a + bi = Im z = r cis( θ ) r cis( − θ ) r cis( θ ) = r θ ccw Re θ cw or − θ ccw z = r cis( − θ ) z

Useful properties z + w = z + w z − w = z − w and w ) = z z ( zw = z w if w ≠ 0 and w z = z Why? Using rectangular form, z z = | z | 2 (a+bi)(a-bi) = a 2 - bi + bi + (bi)(-bi) = a 2 + b 2 This is |z| 2 by Pythagorean Theorem.

Useful properties We only used the z + w = z + w z − w = z − w and rectangular explanation w ) = z z ( during lecture. Both are zw = z w if w ≠ 0 and valid, but polar is even w easier! r 2 is literally |z| 2 . z = z Why? Using polar form, z z = | z | 2 (rcisø)(rcis(-ø)) = r 2 cis(ø-ø) = r 2 cis(0) = r 2

Worked example 3 a + bi Write in rectangular form ( ). 2 + 5 i 3 3 2 + 5 i = 2 + 5 i ⋅ 1 This is the same as 2 + 5 i ⋅ 2 − 5 i 3 multiplying by 1. It does = not change the value of 2 − 5 i the original number. = 3(2 − 5 i ) 2 2 + 5 2 = 6 − 15 i 29 + − 15 6 = 29 i 29

(Multiplicative) Inverses z = a + bi z = r cis θ Given that or that , what are 
 z − 1 = 1 formulas for ? z ( a + bi ) − 1 = a − bi a 2 + b 2 ( r cis θ ) − 1 = ⋯ First, let’ s talk about positive powers in polar form.

Powers z 2 z 15 z 100 What is or or ? Remember: ( r cis θ ) n cis( ) = ___ ___ ? ? ( r cis θ )( s cis ϕ ) = ( rs )cis( θ + ϕ ) 3 z z 1/2 = z 1/3 = z What about or ? (rcisØ) 2 = (rcisØ)(rcisØ) = (rr)cis(Ø+Ø) = r 2 cis(2Ø)

Powers z 2 z 15 z 100 What is or or ? Remember: ( r cis θ ) n cis( ) = ___ ___ ? ? ( r cis θ )( s cis ϕ ) = ( rs )cis( θ + ϕ ) 3 z z 1/2 = z 1/3 = z What about or ? (rcisØ) 3 = (rcisØ) 2 (rcisØ) = (r 2 cis(2Ø))(rcisØ) = (r 2 r)cis(2Ø+Ø) = r 3 cis(3Ø)

Powers z 2 z 15 z 100 What is or or ? Remember: ( r cis θ ) n cis( ) = ___ ___ ? ? ( r cis θ )( s cis ϕ ) = ( rs )cis( θ + ϕ ) 3 z z 1/2 = z 1/3 = z What about or ? (rcisØ) 2 = r 2 cis(2Ø) (rcisØ) 3 = r 3 cis(3Ø)

Powers z 2 z 15 z 100 What is or or ? Remember: ( r cis θ ) n cis( ) = ___ ___ ? ? ( r cis θ )( s cis ϕ ) = ( rs )cis( θ + ϕ ) 3 z z 1/2 = z 1/3 = z What about or ? de Moivre's Formula n ( r cis θ ) n = r n cis( n θ ) For any integer , .

Powers z 2 z 15 z 100 What is or or ? Remember: ( r cis θ ) n r n cis( ) = ___ ___ n θ ( r cis θ )( s cis ϕ ) = ( rs )cis( θ + ϕ ) 3 z z 1/2 = z 1/3 = z What about or ? ( r cis θ ) − 1 = 1 r cis( − θ )

Powers z 2 z 15 z 100 What is or or ? Remember: ( r cis θ ) n r n cis( ) = ___ ___ n θ ( r cis θ )( s cis ϕ ) = ( rs )cis( θ + ϕ ) 3 z z 1/2 = z 1/3 = z What about or ? z = 5 cis(60 ∘ ) We want a number w Im for which w 2 = 5cis(60°). 5cis(30 ∘ ) r 2 cis(2Ø) = 5cis(60°) Re 5cis(30 ∘ ) − r = 5 and Ø = 30° 5cis( − 120 ∘ ) = r = - 5 and Ø = 30°

Powers z 2 z 15 z 100 What is or or ? Remember: ( r cis θ ) n r n cis( ) = ___ ___ n θ ( r cis θ )( s cis ϕ ) = ( rs )cis( θ + ϕ ) 3 z z 1/2 = z 1/3 = z What about or ? Im z = 5 cis(60 ∘ ) = 5 cis(420 ∘ ) Can we use r = 5 and Ø = 210°? 5cis(30 ∘ ) Re 5cis(30 ∘ ) − Yes, 5cis(210°) = 5cis(-120°). 5cis( − 120 ∘ ) = 5cis(210 ∘ ) =

Square roots x x For a positive real number , we require to be positive. − x = i − x x For a negative real number , we say that . − 180 ∘ < θ ≤ 180 ∘ z = r cis θ For a complex number with we 
 r )cis( θ z = ( 2 ) say that .