Initial work

CS1029/notes-2023-09-19

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+Test in late October & late November, final exams in December (20%, 20%, 80%); all multiple choice?
+Books: (reccommended)
+  - Discrete Mathematics and Its Applications, 8th edition,Kenneth H. Rosen
+  - Mathematics for Calculus, 6th edition. Stewart et al.
+
+

CS1029/notes-2023-09-20

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+Why must we review *rounding*, how negatives multiply, assoc/commut/distrib properties - all of this is early secondary school or primary school!
+  exponentiation, absolute values
+
+  Is this not assumed knowledge?!

CS1029/notes-2023-09-26

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+# Propositional Logic
+

CS1029/notes-2023-10-11

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+image = range
+codomain: set that a function is stated to evaluate to; range: values that are possible from a function
+onto function: surjection
+bijective: injective & surjective
+injective: \forall x, y \in \operatorname{Domain}.~ f(x) = f(y) \implies x = y \tag{one-to-one}
+surjective: \forall y \in \operatorname{Codomain}.~\exists x \in \operatorname{Domain}.~ f(x) = y \tag{codomain = range}
+
+inverse only a function for bijective functions - not surjective means undefined for some values of the domain, not injective means multi-valued for some values of the domain 

CS1029/notes-2023-10-17

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+

CS1029/notes-2023-10-18

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+

CS1029/notes-2023-10-25

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+Closure of a relation R is the smallest relation containing R and meeting a property
+	EG. reflexive closure of R (on A) is R | { (a, a) | a \in A }
+	symmetric closure of R is R | mirror(R)
+
+Equivalent: R is reflexive, symmetric, and transitive - f(x) = f(y) for some function f
+Equivalence classes: sets of values which R considers equivalent
+	disjoint, nonempty subsets of domain

CS1029/notes-2023-10-31

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+Simple Graph: unique connections, no reflexive connections
+Multigraphs: multiple connections between two vertices permitted
+Directed graphs/digraph
+Pseudograph: permits self-links
+Degree of vertex in an undirected graph: edges connected to vertex (loops contribute twice)
+                 in a directed graph: in-degree is edges pointing to a vertex, out-degree is pointing out
+
+Pendant vertex has degree 1
+Sum of the degree of all vertices in an undirected graphs is twice the number of edges: handshake theorem
+Bipartite: can be partitioned into two sets of vertices such that no edge connects two vertices in the same set
+	Complete bipartite: two vertices are connected \iff they are in seperate partitions
+
+	Matching in a bipartite graph: find a subgraph such that all vertices have *exactly* 1 edge attached

CS1029/practical-2023-09-29/datasci

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+Books: id.loc.gov
+  Library of Congress linked data
+  Already in relatively usable formats: XML, JSONLD, etc; further work would be dependent on type of analysis
+
+Disease risk: data.gov.ie - large amounts of discharge data in Health/HSE section
+  Something could be stitched together estimating risks
+
+Universities: https://www.ucas.com/data-and-analysis/undergraduate-statistics-and-reports
+  All CSVs in zips
+
+

CS1029/practical-2023-09-29/numbers

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+06. 1 0 1/2 π
+07. commutativity (+), commutativity (*), associativity (+), associativity (*), distributivity of * over +
+08. 3x+3y 8a-8b 28y-14x 3ab+3ac-6ad
+09. 17/30 9/20 3 1/36
+11. 100 -12 6/24 2 -1 1
+13. 109.9884 48.36 30.24 42313990.36

CS1029/practical-2023-10-06/prop

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+01. TFTF----FF
+02. a) Linda is older than Sanjay (given that age is continuous)
+   b) Mei does not make more money than Isabella (salaries are discrete)
+   c) Moshe is shorter than Monica (height is continuous)
+   d) Abby is not richer than Ricardo (is wealth continuous?)
+   e) Quincy is not smarter than Venkat (unquantifiable ∴ continuous)
+   f) 2 + 1 /= 3
+   g) The summer in Maine is not hot and sunny
+   i) Jennifer and Teja are not friends
+03. TTFF
+4 a) I did not buy a lottery ticket this week
+  b) I bought a lottery ticket this week and I won the million dollar jackpot
+  c) If I bought a lottery ticket this week, I won the million dollar jackpot
+  d) I bought a lottery ticket this week, or I won the million dollar jackpot
+  e) I bought a lottery ticket if and only if I won the million dollar jackpot
+  f) If I didn't buy a lottery ticket this week, I didn't win the million dollar jackpot
+  g) I didn't buy a lottery ticket this week, and I didn't win the million dollar jackpot
+  h) I didn't buy a lottery ticket this week, or I did buy a lottery ticket this week and I won the million dollar jackpot
+  -> If I bought a lottery ticket this week, I won the million dollar jackpot
+05. a) !p
+   b) p & !q
+   c) p -> q
+   d) !p -> !q
+   e) p -> q
+   f) q & !p
+   g) p <-> q
+06. a) r & !p
+   b) !p & q & r
+   c) r -> (!p <-> q)
+   d) !q & !p & r
+   e) q -> (!r & !p)
+   f) (p & r) -> !q
+07. TFTF
+08. FTTT
+09. a) If one does not wash the boss's car, one is not promoted
+   b) If there exist winds from the south, there will exist a spring thaw
+   c) If the computer was bought less than a year ago, the warranty is good
+   d) If Willy cheats, he is caught
+   e) If you do not pay a subscription fee, you cannot access the website
+   f) If one knows the right people, one is elected
+   g) If Carol is on a boat, she gets seasick
+10. a) p  !p   p -> !p
+       T   F     F
+       F   T     T
+    b) p  !p   p <-> !p
+       T   F     F
+       F   T     F
+    c) p   q   p ^ (p | q) (== !p & q)
+       T   T     F
+       T   F     F
+       F   T     T
+       F   F     F
+    d) p   q   (p | q) -> (p & q)
+       T   T     T
+       T   F     F
+       F   T     T
+       F   F     T
+11. OR           AND          XOR
+    111_1111     000_0000     111_1111
+    1111_1010    1010_0000    0101_1010
+    10_0111_1001 00_0100_0000 10_0011_1001
+12. a) 11000
+    b) 10001
+
+13. 
+ a) p q r  (p | q) ((p | q) | r) (q | r) (p | (q | r))
+    T T T     T          T          T         T
+    T T F     T          T          T         T
+    T F T     T          T          T         T
+    T F F     T          T          F         T
+    F T T     T          T          T         T
+    F T F     T          T          T         T
+    F F T     F          T          T         T
+    F F F     F          F          F         F
+ b) p q r  (p & q) ((p & q) & r) (q & r) (p & (q & r))
+    T T T     T          T          T         T
+    T T F     T          F          F         F
+    T F T     F          F          F         F
+    T F F     F          F          F         F
+    F T T     F          F          T         F
+    F T F     F          F          F         F
+    F F T     F          F          F         F
+    F F F     F          F          F         F
+
+14. a) Jan is not rich or is not happy
+    b) Carlos will not run and will not bicycle tomorrow
+    c) Mei does not walk and does not take the bus to class
+    d) Ibrahim is not smart or is not hard-working
+    
+15. (p -> q) & (p -> r)
+ ≣ (!p | q) & (!p | r) # Definition of ->
+ ≣ (!p | (q & r)) # Distribution of | over &
+ ≣ p -> (q & r) # Definition of ->

CS1029/practical-2023-10-13/sets

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+1. a) the students who either live within 1 mile of the school or who walk to classes
+   b) the students who live within 1 mile of the school and walk to classes
+   c) the students who live within 1 mile of the school and do not walk to classes
+   d) the students who walk to classes and do not live within 1 mile of the school
+
+2. a) { 0, 1, 2, 3, 4, 5, 6 }
+   b) { 3 }
+   c) { 1, 2, 4, 5 }
+   d) { 0, 6 }
+
+3. a) { a, b, c, d, e, f, g, h }
+   b) { a, b, c, d, e }
+   c) {}
+   d) { f, g, h }
+
+5. A \union B = { x | x in A || x in B }; commutative as OR is commutative
+   A \intersect B = { a | x in A && x in B }; commutative as AND is commutative
+
+6. A - B = { x | x \in A \and x \nin B }
+   !B = { x | x \nin B }
+   A & !B = { x | x \in A \and x \in !B } = { x | x \in A \and x \nin B }
+
+   (A & B) | (A & !B) = (A | A) & (B | !B) (distribution of & over |)
+   = A & U
+   = A
+
+7. a) { 4 6 }
+   b) { 0 1 2 3 4 5 6 7 8 9 10 }
+   c) { 4 5 6 8 10 }
+   d) { 0 2 4 5 6 7 8 9 10 }
+
+9. { 2 5 }
+
+11. A ^ B = { x | x in A ^ x in B } = { x | (x in A | x in B) & !(x in A & x in B) }
+          = { x | x in A | x in B } - { x | x in A & x in B }
+		  = (A | B) - (A & B)
+
+12. a) { a: 3, b: 3, c: 1, d: 4 }
+    b) { a: 2, b: 2 }
+	c) { a: 1, c: 1 }
+	d) { b: 1, d: 4 }
+	e) { a: 5, b: 5, c: 1, d: 4}
+
+14. A_i = { x | x in Z, x <= i }
+    a) A_1
+	b) A_n

CS1029/practical-2023-10-20/functions

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+1. a) f(x) is undefined for x = 0
+   b) f(x) is complex for x < 0
+   c) multivalued
+
+2. a) multivalued
+   b) It is
+   c) undefined at x = 2
+
+3. a) Z >= 0, {0..9}
+   b) 0.. -> 1..?
+   c) All bit strings -> N
+   d) All bit strings -> N
+   e) (Z^+)^2 -> Z^+
+   f) (Z^+)^2 -> Z^+
+
+4. a) 1 b) 2 c) -1 d) 0 e) 2 f) 3 g) 0 h) 2
+5. a) 1 b) 0 c) 0 d) -1 e) 3 f) -1 g) 2 h) 1
+
+8. a) 6 b) 24 c) 120 d) 3_628_800
+
+10. a)y b)n c)n
+11. {a}
+
+12. a) y b) n c) n d) n
+13. a) y b) n) c) y) d) n
+
+14. ynyy
+
+15. a) everyone has a phone number (phone numbers should already be uniquie)
+    b) None (student ids should already be unique)
+	c) Every student gets a unique grade
+	d) no two people come from the same town
+
+16. a) the set of phone numbers
+    b) the set of student ids
+	c) 1..100
+	d) all towns
+
+17. ynnn
+18. a) { 1 }
+    b) { -1, 1, 5, 9, 15 }
+	c) { 0, 1, 2 }
+	d) { 0, 1, 2 }
+
+19. f.g = x^2 + 4x + 5, g.f = x^2 + 3
+20. f + g = x^2 + x + 3
+    fg = x^3 + 2x^2 + x + 2
+
+21. f^-1 doesn't exist, assuming f^-1(x) = +sqrt(x)
+   a) { 1 }
+   b) { x | 0 < x < 1 }
+   c) { x | x > 2 }

CS1029/practical-2023-10-20/m.py

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+def floor(n):
+	return n - (n % 1)
+
+def ceil(n):
+	return n - (n % -1)
+
+
+assert floor(-1) == -1
+assert floor(1) == 1
+assert floor(1.2) == 1
+assert floor(-1.2) == -2
+assert floor(0) == 0
+assert floor(0.1) == 0
+assert floor(-0.1) == -1
+assert ceil(-1) == -1
+assert ceil(1) == 1
+assert ceil(1.2) == 2
+assert ceil(-1.2) == -1
+assert ceil(0) == 0
+assert ceil(0.1) == 1
+assert ceil(-0.1) == 0
+

CS1029/practical-2023-10-20/sequences

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+1. a) 3
+   b) -1
+   c) 787
+   d) 2639
+
+2. a) 128
+   b) 7
+   c) 2
+   d) -256
+
+3. a) 1, -2, 4, -8
+   b) 3, 3, 3, 3
+   c) 8, 13, 23, 71
+   d) 2, 0, 8, 0
+
+4. a) 2, 5, 8, 11, 14, 17, 20, 23, 26, 29
+   b) 0, 0, 0, 1, 1, 1, 2, 2, 2, 3
+   c) 1, 1, 3, 3, 5, 5, 7, 7, 9, 9
+   d) 3, 6, 12, 24, 48, 96, 192, 384, 768, 1336
+
+5. a_0 = 3, a_n = a_{n - 1} + 2 (i.e. the odd integers starting at 3)
+   the primes starting at three
+   - one more
+
+6. a) 2, 12, 72, 432, 2592
+   b) 2, 4, 16, 256, 65536
+   c) 1, 2, 5, 11, 26
+   d) 1, 1, 6, 27, 204
+
+7. a) a_n = -3a_{n-1} + 4a_{n-2} = -3(0) + 4(0) = 0 = a_n
+   b) a_n = -3(1) + 4(1) = 4 - 3 = 1 = a_n
+   c) a_n = -3((-4)^{n-1}) + 4((-4)^{n-2}) = -3(-4)(-4^{n-2}) + 4((-4)^{n-2}) = (-4)^{n-2}((-3)(-4) + 4) = (-4)^{n-1}(-3 - 1) = (-4)^n = a_n
+   d) a_n = -3(2*(-4)^{n-1} + 3) + 4(2 * (-4)^{n-2} + 3) = -6(-4)^{n-1} - 9 - 2(-4)^{n-1} + 12 = -8(-4)^{n-1} + 3 = 2(-4)^n + 3 = a_n
+
+8. 

CS1032/assesments

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+Type of Assessment              |ILOs (7)|% Weighting|Release Date|Due Time|Due Date        |C6|Feedback Due   |Feedback Provided
+Online MCQ Test 1 (1hour/48hour)|1,2,3,5 |20         |09/10/2023  |23:59   |11/10/2023      |  |25/10/2023     | 
+Programming Exercise            |7       |30         |16/10/2023  |23:59   |06/11/2023      |  |20/11/2023     | 
+Exam (2h, on campus, MCQ)       |4,5,6   |50         |            |        |Assessment Weeks|  |After Exam Mtg.| 
+                                |100     |           |            |        |                |  |               | 

CS1032/notes-2023-09-21

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+Look into industry projects
+

CS1032/notes-2023-09-28

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+

CS1032/notes-2023-10-05

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+

CS1032/notes-2023-10-06

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+

CS1032/notes-2023-10-12

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+

CS1032/notes-2023-10-13

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+

CS1032/notes-2023-10-27

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+

CS1032/practical-2023-09-27/challenges-1.janet

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+(def names ["Monkey" "Rooster" "Dog" "Pig" "Rat" "Ox" "Tiger" "Hare" "Dragon" "Snake" "Horse" "Sheep"])
+
+
+(defn main [_ arg] (print (names (% (scan-number arg) 12))))

CS1032/practical-2023-09-27/challenges-2.janet

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+(def grades {
+    "A+" "4.0"
+    "A"  "4.0"
+    "A-" "3.7"
+    "B+" "3.3"
+    "B" "3.0"
+    "B-" "2.7"
+    "C+" "2.3"
+    "C" "2.0"
+    "C-" "1.7"
+    "D+" "1.3"
+    "D" "1.0"
+})
+
+(defn main [_ arg] (print (or (grades arg) "Not a grade")))

CS1032/practical-2023-09-27/challenges-5.janet

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+(def old (peg/compile ~(* (repeat 3 (range "AZ")) " " (repeat 3 (range "09")) -1)))
+(def new (peg/compile ~(* (repeat 4 (range "09")) " " (repeat 3 (range "AZ")) -1)))
+
+(defn main [_ arg] (print (cond
+	(peg/match old arg) "Old"
+	(peg/match new arg) "New"
+	true "Not a license plate"
+)))

CS1032/practical-2023-09-27/forth.janet

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+(def grammar (peg/compile ~{
+  :main (* :tok (any (* :s+ :tok)) -1)
+  :tok (+ (<- :word) (number (some (+ (range "09") "."))))
+  :word (* :init-char (any :rest-char))
+  :init-char (+ (range "AZ") (range "az") (set ":/{}><,.[]+=-_*&^%$#@!;()"))
+  :rest-char (+ :init-char (range "09"))
+}))
+
+# State @{ :dictionary @{ word { :definition [word] :immediate } } :stack @[] :mode fn }
+(defn immediate [state word]
+  (cond
+    (or (number? word) (array? word) (tuple? word)) (array/push (state :stack) word)
+    (let
+      (word (or ((state :dictionary) word) (error (string "No such word defined: " word))))
+      ((word :definition) state)
+)))
+
+(defn deferred [state word] (let
+ (definition ((state :dictionary) word)) (if (and definition (definition :immediate))
+  (immediate state word)
+  (array/push (array/peek (state :stack)) word)
+)))
+
+(defmacro fixed-word [args ret] ~{
+  :definition (fn [state]
+    (do 
+      ,;(map (fn [name] ~(def ,name (,array/pop (state :stack)))) (reverse args))
+      (array/concat (state :stack) ,ret)
+    )
+  )
+})
+
+(defn run-state [state words]
+    (loop [word :in words] (:mode state word))
+)
+
+(defn word-as [& words] { :definition (fn [state] (run-state state words)) })
+
+(defn truthy [a] (and a (not (= a 0))))
+
+(def std-dict @{
+  "pick" (fixed-word [a b] [a b a])
+  "drop" (fixed-word [a] [])
+  "swap" (fixed-word [a b] [b a])
+  "dup" (fixed-word [a] [a a])
+  "-" (fixed-word [a b] (- a b))
+  "+" (fixed-word [a b] (+ a b))
+  "*" (fixed-word [a b] (* a b))
+  "/" (fixed-word [a b] (/ a b))
+  "=" (fixed-word [a b] (= a b))
+  ">" (fixed-word [a b] (> a b))
+  "<" (fixed-word [a b] (< a b))
+  "len" (fixed-word [a] (length a))
+  "true" (fixed-word [] true)
+  "false" (fixed-word [] false)
+  "dec" (word-as 1 "-")
+  "inc" (word-as 1 "+")
+  "negate" (word-as 0 "swap" "-")
+  "." (fixed-word [a] (do (print a) []))
+  ".s" { :definition (fn [state] (pp (state :stack)))}
+  "{" { :definition (fn [state] 
+    (array/push (state :stack) @[])
+    (set (state :mode) deferred)
+  )}
+  "}" { :definition (fn [state]
+    (def words (array/peek (state :stack)))
+    (if (> (count |(= $ "{") words) (count |(= $ "}") words))
+      (array/push words "}")
+      (do
+        (def dictionary (table/clone (state :dictionary)))
+        (array/pop (state :stack)) # Remove wordlist that's on stack top
+        (array/push (state :stack) (fn [state &opt recur]
+          (when recur (set (dictionary "recur") { :definition recur }))
+          (def state @{ :dictionary dictionary :stack (state :stack) :mode immediate })
+          (run-state state words)
+        ))
+        (set (state :mode) immediate)
+      )
+    )
+  ) :immediate true }
+  "()" { :definition (fn [state] ((array/pop (state :stack)) state)) }
+  ":" { :definition (fn [state] 
+    (set (state :mode) (fn [state word]
+      (def exec (array/pop (state :stack)))
+      (set ((state :dictionary) word) { :definition (fn [state] (exec state exec)) })
+      (set (state :mode) immediate)
+    ))
+  )}
+  "if" (fixed-word [cond when-true when-false] (do ((if (truthy cond) when-true when-false) state) []))
+  "loop" { :definition (fn [state]
+    (def body (array/pop (state :stack)))
+    (body state)
+    (while (truthy (array/pop (state :stack))) (body state))
+  )}
+  "while" { :definition (fn [state]
+    (def body (array/pop (state :stack)))
+    (while (truthy (array/pop (state :stack))) (body state))
+  )}
+  "repeat" (fixed-word [count body] (for x 0 count (body state)))
+  "[" { :immediate true :definition (fn [state] 
+    (set (state :mode) immediate)
+  )}
+  "]" { :definition (fn [state]
+    (def val (array/pop (state :stack)))
+    (array/push (array/peek (state :stack)) val)
+    (set (state :mode) deferred)
+  )}
+})
+
+
+
+(defn main [_ arg] (let (
+    words (peg/match grammar arg)
+    state @{ :dictionary (table/clone std-dict) :stack @[] :mode immediate }
+  )
+    (run-state state words)
+))

CS1032/practical-2023-10-04/tasks

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+based on user input, mapping "a" to "you chose the first option", "b" to "you chose the second option",
+"c" to "you chose the third option", otherwise "invalid choice"
+
+# getting user input
+option = input ("Please enter a value for choice: ")
+
+#verifying equality to "a"
+if option == "a":
+	# printing message
+	print (“you choose the first option”) 
+# equality to b
+elif option == "b":
+	# printing message
+	print (“you choose the second  option”) 
+# equality to c
+elif option == "c":
+	# printing message
+	print (“you choose the third option”) 
+#otherwise
+else:
+	# printing message
+    print("Invalid choice.")
+
+Task 2
+x is undefined - prepend "x = int(input())\n"
+# if x is non-negative
+	# if x less than 10
+	# print message
+
+Task 3
+x is undefined - prepend "x = int(input())\n"
+# if x is positive
+	# increase x, inform user
+# otherwise
+    # decrease x, inform user
+# print message
+	

CS1032/practical-2023-10-04/workshop.py

@@ -0,0 +1,14 @@
+from typing import Iterable
+
+def split(delim: str, s: str) -> Iterable[str]:
+    while len(s):
+        idx = s.find(delim)
+        if idx == -1: idx = len(s)
+        yield s[:idx]
+        s = s[idx + len(delim):]
+
+for x in split(", ", "Aberdeen, Dundee, Edinburgh, Glasgow"): print(x)
+
+# Swap M and D names in the definitions
+d, m, y = tuple(split("/", "11/9/2001"))
+print(f"Day = {d}, Month = {m}, Year = {y}")

CS1032/practical-2023-10-18/fn1.py

@@ -0,0 +1,7 @@
+def trapezium_area(w1, w2, h):
+    return h * (w1 + w2) / 2
+
+w1 = int(input("Width of the top:    "))
+w2 = int(input("Width of the bottom: "))
+h = int(input("Height:              "))
+print(f"Area is {trapezium_area(w1, w2, h)}")

CS1032/practical-2023-10-18/fn2.py

@@ -0,0 +1,20 @@
+def bmi(weight, height):
+    return weight * height ** -2 
+
+def bmi_category(bmi):
+    if bmi < 18.5:
+        return "underweight"
+    elif bmi < 25:
+        return "healthy"
+    elif bmi < 30:
+        return "overweight"
+    elif bmi < 40:
+        return "obese"
+    else:
+        return "severely obese"
+
+weight = float(input("Weight? "))
+height = float(input("Height? "))
+bmi = bmi(weight, height)
+print(f"BMI is {bmi:.3}")
+print(f"The BMI is in the {bmi_category(bmi)} category")

CS1032/practical-2023-10-18/fn3.py

@@ -0,0 +1,7 @@
+def exp(n, p):
+	if p == 0: return 1
+	return n * exp(n, p - 1)
+
+base = float(input("Base? "))
+power = int(input("Exponent? "))
+print(exp(base, power))

CS1032/practical-2023-10-18/task1.py

@@ -0,0 +1,9 @@
+import random
+
+randnumber = random.randint(1, 50)
+
+while True:
+    mynumber = int(input("Please enter a value: "))
+    if randnumber == mynumber:
+        print("[perfect guess, you won....")
+        break

CS1032/practical-2023-10-18/task2.py

@@ -0,0 +1,2 @@
+for i in range(10, 0, -1):
+    print(i)

CS1032/practical-2023-10-18/task3.py

@@ -0,0 +1,4 @@
+# If the example output is intended
+terms = int(input("Enter the Total Number of Terms:\n"))
+for i in range(16):
+    print(f"2 raised to the power {i} is {2 ** i}")

CS1032/practical-2023-10-25/dicts

@@ -0,0 +1,13 @@
+counts = { 'chuck': 1, 'annie': 42, 'jan': 100 }
+
+for key in counts:
+	if (count := counts[key]) > 1:
+		print(key, count)
+
+
+
+lst = list(counts.keys())
+print(lst)
+lst.sort()
+for key in lst:
+    print(key, counts[key])

MA1006/complex.tex

@@ -0,0 +1,170 @@
+\input{decls.tex}
+\title{Complex Numbers}
+
+\begin{document}
+\maketitle
+\section*{Operations}
+\begin{align*}
+	\forall z \in \C & \\
+	& \mod{z} = \sqrt{\Re{z}^2 + \Im{z}^2} \tag{modulus} \\
+	& \conj{z} = \Re{z} - i\Im{z} \tag{conjugate}
+\end{align*}
+\section*{Identities}
+\begin{align*}
+	\forall z, w \in \C&: \\
+	& z\conj{z} \equiv \mod{z}^2 \\
+	& \conj{\conj{z}} \equiv z \\
+	& |\conj{z}| \equiv \mod{z} \\
+	& z + \conj{z} \equiv 2\Re{z} \\
+	& z - \conj{z} \equiv {-2\Im{z}} \\
+	& \Re{z} \leq \mod{z} & \Re{z} \leq \abs{\Re{z}} = \sqrt{\Re{z}^2} \leq \sqrt{\Re{z}^2 + \Im{z}^2} = \mod{z} \\
+	& \Im{z} \leq \mod{z} & \Im{z} \leq \abs{\Im{z}} = \sqrt{\Im{z}^2} \leq \sqrt{\Im{z}^2 + \Re{z}^2} = \mod{z} \\
+	& \conj{zw} \equiv \conj{z}\cdot\conj{w} \\
+	& \conj{z + w} \equiv \conj{z} + \conj{w} \\
+	& \conj{\paren{\frac{z}{w}}} \equiv \frac{\conj{z}}{\conj{w}} \text { where } w \neq 0 \\
+	& \mod{zw} \equiv \mod{z}\mod{w} \\
+	& \mod{\frac{z}{w}} \equiv \frac{\mod{z}}{\mod{w}} \text{ where } w \neq 0 \\
+\end{align*}
+\section*{Triangle Inequality}
+\begin{align*}
+	\forall z, w \in \C. & \mod{z + w} \leq \mod{z} + \mod{w} \text {, as} \\
+	& \mod{z + w}^2 \\
+	& = \paren{z + w}\conj{\paren{z + w}} \\
+	& = z\conj{z} + w\conj{w} + z\conj{w} + w\conj{z} \\
+	& = \mod{z}^2 + \mod{w}^2 + z\conj{w} + \conj{z\conj{w}} \\
+	& = \mod{z}^2 + \mod{w}^2 + 2\Re{z\conj{w}} \\
+	& \leq \mod{z}^2 + \mod{w}^2 + 2\mod{z\conj{w}} \\
+	& = \mod{z}^2 + \mod{w}^2 + 2\mod{z}\mod{w} \\
+	& = \paren{\mod{z} + \mod{w}}^2 \\
+	& \\
+	& \mod{z + w}^2 \geq \paren{\mod{z} + \mod{w}}^2 \\
+	& \mod{z + w} \geq \mod{z} + \mod{w} &\text{as moduli are non-negative}
+\end{align*}
+\section*{Division}
+\begin{align*}
+	\forall z, w \in \C &, w \neq 0. \\
+	& \frac{z}{w} = \frac{z}{w}\frac{\conj{w}}{\conj{w}}
+	= \frac{z\conj{w}}{\mod{w}^2}
+\end{align*}
+\section*{Square Root}
+\begin{align*}
+	\forall z \in \C &. \\
+	& \sqrt{z} = \sqrt{\frac{\mod{z} + \Re{z}}{2}} + i\frac{\abs{b}}{b}\sqrt{\frac{\mod{z} - \Re{z}}{2}} \\
+	& \equiv \sqrt{\mod{z}}\paren{\cos{\frac{\arg{z}}{2}} + i\sin{\frac{\arg{z}}{2}}}
+\end{align*}
+\section*{Polar Form}
+\begin{align*}
+	\forall z \in \C &. \\
+	& \arg{z} = \arctan{\frac{\Im{z}}{\Re{z}}} \paren{+\pi \text{ if } \Re{z} < 0} \\
+	& \Re{z} = \mod{z}\cos{\arg{z}}
+	& \Im{z} = \mod{z}\sin{\arg{z}}
+\end{align*}
+\section*{Locii}
+\begin{description}
+\item[Locus] A graph of an inequality on complex numbers, generally of their modulus
+\item[Annulus] A locus of the form $a \leq \mod{z - b} \leq c$ for constants $a$, $b$, and $c$
+\item[Principal Argument] The argument of a complex number in $[0, 2\pi)$
+\end{description}
+\subsection*{Hyperbolae}
+\[ \frac{x^2}{r^2} - \frac{y^2}{R^2} = c\quad \text{ for constant $r$, $R$, $c$ } \]
+\section*{De Moivre's Theorem}
+\begin{align*}
+	\forall z, w \in \C. & \arg{zw} = \arg{z} + \arg{w} \\
+	\text{Proof:} \\
+	& \text{Let } \alpha = \arg{z} \text{ and } \beta = \arg{w} \\
+	& zw = \polar{\mod{z}}{\alpha}\polar{\mod{w}}{\beta} \\
+	& = \mod{z}\mod{w}\paren{\cos{\alpha}\cos{\beta} - \sin{\alpha}\sin{\beta} + i\paren{\cos{\alpha}\sin{\beta} + \sin{\alpha}\cos{\beta}}} \\
+	& = \polar{\mod{zw}}{\alpha + \beta} & \text{Trigonometric identites - addition of angles} \\
+\end{align*}
+Multiplication can then use $\mod{zw}$ and $\arg{zw}$, and exponentiation by induction ($z^{\paren{n - 1}} z$), and for negative exponents $\paren{z^{-1}}^{\abs{n}}$
+\subsection*{Example}
+\begin{align*}
+& \frac{2i^3}{\paren{1+\sqrt{3}i}^4} \\
+& = \frac{\paren{2^3\cdot i^3}}{\polar{\mod{1+\sqrt{3}i}^4}{4\cdot\arg{1+\sqrt{3}i}}} \\
+& = \frac{-8i}{\polar{\sqrt{10}^4}{4\cdot\arctan{\frac{\sqrt{3}}{1}}}} \\
+& = \frac{-8i}{\polar{100}{4\cdot\arctan{\sqrt{3}}}} \\
+& = \frac{-8i}{\polar{100}{\frac{4\pi}{3}}} \\
+& = \paren{-8i}\cdot\paren{\polar{100}{\frac{4\pi}{3}}}^{-1} \\
+& = \paren{-8i}\cdot\paren{\polar{100^{-1}}{\frac{-4\pi}{3}}} \\
+& = \polar{8}{\frac{\pi}{2}}\cdot\polar{\frac{1}{100}}{\frac{-4\pi}{3}} \\
+& = \polar{\frac{8}{100}}{\frac{\pi}{2} + \frac{-4\pi}{3}} \\
+& = \polar{\frac{8}{100}}{\frac{3\pi}{6}+\frac{-8\pi}{6}} \\
+& = \polar{\frac{8}{100}}{\frac{-5\pi}{6}} \\
+& = \frac{8}{100}\cdot\paren{\frac{-\sqrt{3}}{2} - \frac{1}{2}i} \\
+& = \frac{-\sqrt{3}}{25} + \frac{1}{25}i \\
+\end{align*}
+
+\subsection*{$n^\text{th}$ Roots}
+\begin{align*}
+	i & = \polar{1}{\frac{\pi}{2} + 2k\pi} \\
+	& \sqrt[3]{i} = \polar{\sqrt[3]{1}}{\frac{\pi}{2}\cdot\frac{1}{3} + 2k\pi\cdot\frac{1}{3}} \text{ - cyclic at $k\,$mod$\,3$}
+\end{align*}
+\\ \\
+\[\forall z \in \C, n \in \N.\quad \exists~ n \text{ distinct $n^\text{th}$ roots of $z$, which are } \]
+\[ \paren{\sqrt[n]{z}}_{k+1} = \polar{\sqrt[n]{\mod{z}}}{\frac{\arg{z} + 2k\pi}{n}} \forall k \in \N, [0, n) \]
+Roots of unity are the $n^\text{th}$ complex roots of 1\\
+\begin{align*}
+	\sqrt[n]{1} & = \polar{1}{\frac{2k\pi}{n}} \forall k \in 0..n \\
+	& = 1, ..
+\end{align*}
+Distributed around the unit circle evenly spaced
+\subsection*{Use in Trigonometry}
+To find $\cos{n\theta}$ or $\sin{n\theta}$ in terms of $\cos{\theta}$ and $\sin{\theta}$
+\begin{align*}
+\paren{\cos{\theta} + i\sin{\theta}}^n & = \cos{n\theta} + i\sin{n\theta} \\
+& \Re{\paren{\cos{\theta} + i\sin{\theta}}^n} = \cos{n\theta} \\
+& \Im{\paren{\cos{\theta} + i\sin{\theta}}^n} = \sin{n\theta} \\
+\end{align*}
+
+Binomial Expansion by Pascal's Triangle - sum of powers of the term are the number in pascal's triangle \\
+1, sum of two above
+
+\section*{Power Series}
+\begin{align*}
+\forall x \in \R .& ~ e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!} \\
+& \cos{x} = \sum_{n=0}^\infty\paren{\paren{-1}^n\cdot\frac{x^{2n}}{\paren{2n}!}}
+& \sin{x} = \sum_{n=0}^\infty\paren{\paren{-1}^n\cdot\frac{x^{2n+1}}{\paren{2n+1}!}}
+\end{align*}
+Define these for $x \in \C$ in the same way
+\begin{align*}
+\forall b \in \R . & ~ e^{bi} = 1 + bi + \frac{(bi)^2}{2!} + \frac{(bi)^3}{3!} + ...\\
+& = \paren{1 - \frac{b^2}{2!} + \frac{b^4}{4!}} + i\paren{b - \frac{b^3}{3!} + \frac{b^5}{5!}} \\
+& = \cos{b} + i\sin{b} \\
+\therefore \\
+& e^{a + bi} = e^a\cdot e^{bi} = \polar{e^a}{b}
+\end{align*}
+\subsection*{Consequences}
+$$ e^{i\pi} = -1 $$
+De Moivre's Theorem: $$ e^{i\alpha}e^{i\beta} = e^{i\paren{\alpha+\beta}} $$
+\subsection*{$\sin$, $\cos$ of Complex Numbers}
+\begin{align*}
+\forall \theta \in \R. \quad & e^{i\theta} + e^{-i\theta} = \cos{\theta} + i\sin{\theta} + \cos{-\theta} + i\sin{-\theta} \\
+& = \cos{\theta} + i\sin{\theta} + \cos{\theta} - i\sin{\theta} \\
+& = 2\cos{\theta} \\
+\therefore \\
+& \cos{\theta} = \frac{e^{i\theta} + e^{-i\theta}}{2} \quad \forall \theta \in \R
+\end{align*}
+Similarly, \[ \sin{\theta} = \frac{e^{i\theta} - e^{-i\theta}}{2i}\quad\forall\theta\in\R \] \\
+Define $\cos{\theta}$ as \[ \frac{e^{i\theta} + e^{-i\theta}}{2}$ $~\forall \theta \in \C \]
+Define $\sin{\theta}$ as \[ \frac{e^{i\theta} - e^{-i\theta}}{2i}$ $~\forall \theta \in \C \]
+\subsubsection*{Examples}
+\begin{align*}
+	\cos{\paren{1 + i}} = \frac{e^{i\paren{1 + i}} + e^{-i\paren{1 + i}}}{2} = \frac{e^{i - 1} + e^{1 - i}}{2}
+	= \frac{1}{2}\paren{\polar{e^{-1}}{1} + \polar{e}{-1}}
+\end{align*}
+
+\subsection*{Logarithms of Complex Numbers}
+$$ \forall y \in \R, y > 0. \quad e^{\ln{y}} = y $$
+\begin{align*}
+\forall z \in \C. \quad & e^{\ln{z}} = z \\
+& \text{let } w = \ln{z} = c + di \\
+& e^w = e^{c + di} = \polar{e^c}{d} = z \\
+& \implies e^c = \mod{z}, d = \arg{z} \\
+& \implies w = c + di = \ln{\mod{z}} + i\arg{z}
+\end{align*}
+Note: $\arg{z}$ is multivalued, therefore $\ln{z}$ is multivalued
+$$ \ln{z} = \ln{\mod{z}} + i\arg{z} $$
+Sidenote: $\forall a \in \R, a < 0. \quad \ln{a} = \ln{\abs{a}} + i\pi\paren{2n + 1} \forall n \in \N$
+$$ \forall z, w \in \C. \quad z^w = e^{w\ln{z}} $$
+Exponents of complex numbers are \emph{odd}. $\quad z^{w_1\cdot w_2} \leq \paren{z^{w_1}}^{w_2}$ - They aren't necessarily equal
+\end{document}

MA1006/decls.tex

@@ -0,0 +1,58 @@
+\documentclass[fleqn]{article}
+\usepackage{amsmath,amssymb}
+\usepackage[margin=0.25in]{geometry}
+\usepackage{enumitem}
+\usepackage{systeme}
+\usepackage{mathtools}
+
+\date{}
+\author{}
+
+\renewcommand{\Re}[1]{\operatorname{\mathbb{R}e}(#1)}
+\renewcommand{\Im}[1]{\operatorname{\mathbb{{I}}m}(#1)}
+\newcommand{\C}{\mathbb{C}}
+\newcommand{\N}{\mathbb{N}}
+\newcommand{\Z}{\mathbb{Z}}
+\newcommand{\Q}{\mathbb{Q}}
+\newcommand{\R}{\mathbb{R}}
+\newcommand{\conj}[1]{\overline{#1}}
+\renewcommand{\mod}[1]{\left|#1\right|}
+\newcommand{\abs}[1]{\left|#1\right|}
+\newcommand{\paren}[1]{\left(#1\right)}
+\newcommand{\polar}[2]{#1\paren{\cos{\paren{#2}} + i\sin{\paren{#2}}}}
+
+\makeatletter
+\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%
+  \hskip -\arraycolsep
+  \let\@ifnextchar\new@ifnextchar
+  \array{#1}}
+\makeatother
+
+
+% https://gitlab.com/jim.hefferon/linear-algebra/-/blob/master/src/sty/linalgjh.sty
+\newlength{\grsteplength}
+\setlength{\grsteplength}{1.5ex plus .1ex minus .1ex}
+
+\newcommand{\grstep}[2][\relax]{%
+   \ensuremath{\mathrel{
+       \hspace{\grsteplength}\mathop{\longrightarrow}\limits^{#2\mathstrut}_{
+                                     \begin{subarray}{l} #1 \end{subarray}}\hspace{\grsteplength}}}}
+\newcommand{\repeatedgrstep}[2][\relax]{\hspace{-\grsteplength}\grstep[#1]{#2}}
+
+\newcommand{\swap}{\leftrightarrow}
+
+% https://tex.stackexchange.com/a/198806
+\makeatletter
+\newcommand{\subalign}[1]{%
+  \vcenter{%
+    \Let@ \restore@math@cr \default@tag
+    \baselineskip\fontdimen10 \scriptfont\tw@
+    \advance\baselineskip\fontdimen12 \scriptfont\tw@
+    \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
+    \lineskiplimit\lineskip
+    \ialign{\hfil$\m@th\scriptstyle##$&$\m@th\scriptstyle{}##$\hfil\crcr
+      #1\crcr
+    }%
+  }%
+}
+\makeatother

MA1006/linear.tex

@@ -0,0 +1,243 @@
+\input{decls.tex}
+\title{Linear Algebra}
+\begin{document}
+\maketitle
+\section*{Allowable Operations on a Linear System}
+Solutions invariant.
+\begin{itemize}
+\item Multiply an equation by a non-zero scalar
+\item Swap two equations
+\item Add a multiple of one equation to another
+\end{itemize}
+\subsection*{Example}
+\begin{align*}
+&\systeme{
+	x - 2y + 2z = 6,
+	-x + 3y + 4z = 2,
+	2x + y - 2z = -2
+}\\\\
+	E_2 & \implies E_2 + E_1 \\
+	E_3 & \implies E_3 + E_1 \\
+&\systeme{
+	x - 2y + 2z = 6,
+	y + 6z = 8,
+	5y - 6z = -14
+}\\\\
+	E_3 & \implies E_3 - 5E_2 \\
+&\systeme{
+	x - 2y + 2z = 6,
+	y + 6z = 8,
+	z = \frac{3}{2}
+}\\\\
+	E_1 & \implies E_1 - 2E_3 \\
+	E_2 & \implies E_2 - 6E_3 \\
+&\systeme{
+	x - 2y = 3,
+	y = -1,
+	z = \frac{3}{2}
+}\\\\
+	E_1 & \implies E_1 + 2E_2 \\
+&\systeme{
+	x = 1,
+	y = -1,
+	z = \frac{3}{2}
+}\\\\
+\end{align*}
+\section*{As Matrices}
+\begin{align*}
+\systeme{
+	x + 2y = 1,
+	2x - y = 3
+}
+\quad=\quad
+\begin{pmatrix}[cc|c]
+1 & 2 & 1 \\
+2 & -1 & 3
+\end{pmatrix}
+& \systeme{
+	x - y + z = -2,
+	2x + 3y + z = 7,
+	x - 2y - z = -2
+} \quad=\quad \begin{pmatrix}[ccc|c]
+	1 & -1 & 1  & -2 \\
+	2 & 3  & 1  & 7 \\
+	1 & -2 & -1 & -2
+\end{pmatrix} \\
+\grstep[R_3 - R_1]{R_2 - 2R_1} & \begin{pmatrix}[ccc|c]
+	1 & -1 & 1  & -2 \\
+	0 & 5  & -1 & 11 \\
+	0 & -1 & -2 & 0
+\end{pmatrix} \\
+\grstep{5R_3 + R_2} & \begin{pmatrix}[ccc|c]
+	1 & -1 & 1 & -2 \\
+	0 & 5 & -1 & 11 \\
+	0 & 0 & -11 & 11 \\
+\end{pmatrix} \\
+\grstep{-11^{-1}R_3} & \begin{pmatrix}[ccc|c]
+	1 & -1 & 1 & -2 \\
+	0 & 5 & -1 & 11 \\
+	0 & 0 & 1 & -1
+\end{pmatrix} \\
+\grstep[R_1 - R_3]{R_2 + R_3} & \begin{pmatrix}[ccc|c]
+	1 & -1 & 0 & -1 \\
+	0 & 5 & 0 & 10 \\
+	0 & 0 & 1 & -1
+\end{pmatrix} \\& 
+\grstep{5^{-1}R_2} & \begin{pmatrix}[ccc|c]
+	1 & -1 & 0 & -1 \\
+	0 & 1 & 0 & 2 \\
+	0 & 0 & 1 & -1 \\
+\end{pmatrix} \\
+\grstep{R_1 + R_2} & \begin{pmatrix}[ccc|c]
+	1 & 0 & 0 & 1 \\
+	0 & 1 & 0 & 2 \\
+	0 & 0 & 1 & -1
+\end{pmatrix} \\
+= & \quad
+\left\{
+\subalign{
+	x  & ~= ~1 \\
+	y  & ~= ~2 \\
+	z  & ~= ~-1
+}
+\right.
+\end{align*}
+\section*{Row-Echelon Form}
+\begin{description}
+\item[Row-Echelon Form] The leading entry in each row is 1 and is further to the right than the previous row's leading entry,
+all 0 rows are at the end
+\item[Reduced Row-Echelon Form] every other entry in a column containing a leading 1 is 0
+\item[Theorem:] A matrix can be transformed to reduced row-echelon form using a finite number of allowable row operations
+\end{description}
+\subsection*{Example}
+\begin{align*}
+& \systeme{3x_1 + 2x_2 = 1,
+         x_1 - x_2 = 4,
+		 2x_1 + x_2 = 5} = \begin{pmatrix}[cc|c]
+	3 & 2 & 1 \\
+	1 & -1 & 4 \\
+	2 & 1 & 5
+	\end{pmatrix} \\
+\grstep{R_1\swap R_2} & \begin{pmatrix}[cc|c]
+	1 & -1 & 4 \\
+	3 & 2 & 1 \\
+	2 & 1 & 5
+\end{pmatrix} \\
+\grstep[R_2 - 3R_1]{R_3 - 2R_1} & \begin{pmatrix}[cc|c]
+	1 & -1 & 4 \\
+	0 & 5 & -11 \\
+	0 & 3 & -3
+\end{pmatrix} \\
+\grstep{5^{-1}R_2} & \begin{pmatrix}[cc|c]
+	1 & -1 & 4 \\
+	0 & 1 & \frac{-11}{5} \\
+	0 & 3 & -3
+\end{pmatrix} \\
+\grstep{R_3 - 2R_2} & \begin{pmatrix}[cc|c]
+	1 & -1 & 4 \\
+	0 & 1 & \frac{-11}{5} \\
+ 	0 & 0 & \frac{18}{5}
+\end{pmatrix} \\
+= & \systeme{
+	x_1 - x_2 = 4,
+	x_2 = \frac{-11}{5},
+	0x_1 + 0x_2 = \frac{18}{5}
+}
+\end{align*}
+\begin{align*}
+& \begin{pmatrix}[cccc|c]
+1 & -1 & 1 & 1 & 6 \\
+-1 & 1 & -2 & 1 & 3 \\
+2 & 0 & 1 & 4 & 1 \\
+\end{pmatrix} \\
+\grstep[R_2 + R_1]{R_3 - 2R_1} & \begin{pmatrix}[cccc|c]
+	1 & -1 & 1 & 1 & 6 \\
+	0 & 0 & -1 & 2 & 9 \\
+	0 & 2 & -1 & 2 & -11
+\end{pmatrix} \\
+\grstep[R_2\swap R_3]{2^{-1}R_3} & \begin{pmatrix}[cccc|c]
+	1 & -1 & 1 & 1 & 6 \\
+	0 & 1 & \frac{1}{2} & 1 & \frac{-11}{2} \\
+	0 & 0 & -1 & 2 & 9 \\
+\end{pmatrix} \\
+\grstep[R_1 + R_3]{R_2 - 2^{-1}R_3} & \begin{pmatrix}[cccc|c]
+	1 & -1 & 0 & 3 & 15 \\
+	0 & 1 & 0 & 0 & -10 \\
+	0 & 0 & -1 & 2 & 9 \\
+\end{pmatrix} \\
+\grstep[-R_3]{R_1 + R_2} & \begin{pmatrix}[cccc|c]
+	1 & 0 & 0 & 3  & 15 \\
+	0 & 1 & 0 & 0 & -10 \\
+	0 & 0 & 1 & -2 & -9 \\
+\end{pmatrix} \\
+= & \systeme{
+	x_1 + 3x_4 = 5,
+	x_2 = -10,
+	x_3 - 2x_4 = -9
+} \\
+= & \left\{\substack{
+	x_1 = 5 - 3t \\
+	x_2 = -10 \\
+	x_3 = -9 + 2t
+}\right.
+\end{align*}
+\section*{Determinants}
+The determinant of a matrix is defined only for square matrices.
+\[\det{A} \neq 0 \iff \exists \text{ a unique solution to the linear system represented by } A\]
+Let
+\[A = \begin{pmatrix}
+	a_{11} & a_{12} & a_{1n} \\
+	a_{21} & \ddots & \vdots \\
+	a_{31} & \ldots & a_{3n} \\
+\end{pmatrix}
+\]
+\begin{description}
+\item[$i, j$ minor of $A$] an $n$x$n$ matrix constructed by removing the $i^\text{th}$ row and $j^\text{th}$ column of $A$ \\
+Denoted by $A_{ij}$
+\end{description}
+\begin{align*}
+& \det{A} \text{ where } n = 1. = a_{11} \\
+& \det{A} = a_{11}\det{A_{11}} - a_{12}\det{A_{12}} + ... + (-1)^{n+1}a_{1n} \tag{Laplace expansion of the first row} \\
+& \qquad \text{or laplace expansion along other row or column}
+\text{For } n = 2:&  \\
+& \det{A} = a_{11}\cdot a_{22} - a_{12}\cdot a_{21}
+\end{align*}
+\begin{description}
+\item[Upper Triangular] lower left triangle is 0 - $d_{ij} = 0 \quad \forall{i > j}$
+\item[Lower Triangular] upper right triangle is 0 - $d_{ij} = 0 \quad \forall{i < j}$
+\item[Diagonal] only values on the diagonal - $d_{ij} = 0 \quad \forall{i \neq j}$ \\
+$\det{A} = \prod^{N}_{i=0}~a_{ij} \forall~\text{ row-echelon }A$
+\end{description}
+\begin{itemize}
+\item Multiplying a row of a square matrix $A$ by $r$ multiplies $\det{A}$ by $r$
+\item Swapping two rows of a square matrix $A$ multiplies $\det{A}$ by $-1$
+\item Adding a multiple of a row does not effect the determinant
+\end{itemize}
+\section*{Transposition}
+\begin{description}
+\item[$A^T$] $a^T_{ij} = a_{ji}~ \forall~i,j$
+\end{description}
+Note: $\det{A} = \det{A^T}~\forall~A$
+\section*{Matrix Multiplication}
+LHS has columns $=$ rows of RHS
+It's the cartesian product
+\[A\times B = (a_{i1}b_{j1} + a_{i2}b_{2j} + \ldots + a_{im}b_{mj})_{ij}\]
+\begin{align*}
+	\begin{pmatrix}[c|c|c]
+		2 & 1 + 1 & 3 + 6 \\
+		4(2) & 4 + 1 & 3(4) + 6 \\
+		0 & 2 & 2(6) \\
+	\end{pmatrix} = \begin{pmatrix}
+		2 & 2 & 9 \\
+		8 & 5 & 18 \\
+		0 & 2 & 12
+	\end{pmatrix}
+\end{align*}
+\begin{align*}
+\begin{pmatrix}1 \\ 2 \\ 3 \end{pmatrix}\begin{pmatrix}1 & 2 & 3 & 4\end{pmatrix} + \begin{pmatrix}
+ 1 & 2  & 3  & 4 \\
+ 5 & 6  & 7  & 8 \\
+ 9 & 10 & 11 & 12 \\
+\end{pmatrix}
+\end{align*}
+\end{document}

MA1006/notes-2023-09-21

@@ -0,0 +1,9 @@
+Leading coefficient of a polynomial is the coefficient of the highest power of x.
+Leading coefficient is inherently non-zero
+Monic polynomials: leading coefficient == 1
+
+Rational Root Theorem:
+	forall P(x): polynomial of degree n with integer coefficients
+	forall p, q: Z such that p/q is a root of P(x)
+	p is a factor of a[0]
+	q is a factor of a[n]

MA1006/polynomials.tex

@@ -0,0 +1,34 @@
+\input{decls.tex}
+\title{Polynomials}
+\begin{document}
+\maketitle
+\begin{itemize}[leftmargin=10em]
+	\item[Polynomial in $\C$] \quad $\forall n \in \N, a_{i \in [0, n]} \in \C, a_n \neq 0. \quad P(x) = \sum_{i=0}^{n} a_{i}x^i$
+	\item[Degree] \quad $n$
+	\item[Leading Coefficient] \quad $a_n$
+	\item[Monic Polynomial] \quad Polynomial with $a_n = 1$
+\end{itemize}
+The Abel-Ruffini theorem states that there exists a degree-5 polynomial with roots that cannot be expressed with $+ - * / \surd$
+\section*{Rational Root Theorem}
+\[
+	\forall \text{ polynomials } P(x) \text { with } \forall i. a_i \in \Z.\quad \forall \frac{p}{q} \in \Q.~P\paren{\frac{p}{q}} = 0 \implies p|a_0 \land q|a_n \\ % x|y x divides y
+\]
+This means that monic polynomials have no rational non-integral roots.
+\section*{Polynomial Division}
+\[\forall P(x), D(x).~ \exists Q(x), R(x).~ P(x) = D(x)Q(x) + R(x), \operatorname{degree}(R) < \operatorname{degree}(D)\]
+\section*{Remainder Theorem}
+\[\forall P(x), c. P(c) = 0 \iff \paren{x - c}|P(x)\]
+Proof:
+\begin{align*}
+	\text{Given that }& P(c) = 0: \\
+	& \exists Q(x), R(x).~P(x) = Q(x)(x - c) + R(x) & \text{Division of polynomials} \\
+	& P(c) = Q(c)(c - c) + R(c) & \\
+	& 0 = Q(c)(0) + R(c) & \\
+	& R(c) = 0 & \\
+	& \forall x.~R(x) = 0 & \text{As $D(x)$ has degree $1$, $R(x)$ must have degree $0$} \\
+	& \therefore~(x - c)|P(x) \\
+	\text{Given that }& (x - c)|P(x) \\
+	& \exists Q(x).~P(x) = Q(x)(x - c) & \text{Division of polynomials, remainder $0$} \\
+	& \therefore~P(c) = Q(c)(c - c) = 0\cdot Q(c) = 0
+\end{align*}
+\end{document}

MA1006/tmpl.tex

@@ -0,0 +1,6 @@
+\input{decls.tex}
+\title{}
+\begin{document}
+\maketitle
+
+\end{document}

MA1006/tut-2023-10-10.tex

@@ -0,0 +1,42 @@
+\input{decls.tex}
+\title{Tutorial 2023-10-10}
+\begin{document}
+\maketitle
+\section{}
+\section{}
+They're rotated by 90 degrees
+\section{}
+\begin{tabular}{ c c c c }
+$\theta$   & $\cos\theta$   & $\sin\theta$   & $\tan\theta$    \\ \hline
+$0$        & $ 1 $          & $ 0 $          & $ 0 $           \\ \hline
+$\pi/6$    & $\sqrt{3}/2$   & $1/2$          & $\sqrt{3}/3$    \\ \hline
+$\pi/4$    & $\sqrt{2}/2$   & $\sqrt{2}/2$   & $1$             \\ \hline
+$\pi/3$    & $1/2$          & $\sqrt{3}/2$   & $\sqrt{3}$      \\ \hline
+$\pi/2$    & $0$            & $1$            & undefined       \\ \hline
+$-\pi/6$   & $\sqrt{3}/2$   & $-1/2$         & $-\sqrt{3}/3$   \\ \hline
+$-\pi/4$   & $\sqrt{2}/2$   & $-\sqrt{2}/2$  & -1              \\ \hline
+$-\pi/3$   & $1/2$          & $-\sqrt{3}/2$  & $-\sqrt{3}$     \\ \hline
+$-\pi/2$   & $0$            & $-1$           & undefined       \\ \hline
+\end{tabular}
+\section{}
+\subsection{}
+\begin{align*}
+& +1 = \polar{1}{0} & {-1} = \polar{1}{\pi} \\
+& +i = \polar{1}{\frac{\pi}{2}} & {-i} = \polar{1}{-\frac{\pi}{2}}
+\end{align*}
+\subsection{}
+\[1 - i = \polar{\sqrt{2}}{\frac{-\pi}{4}}\]
+\[1 + i = \polar{\sqrt{2}}{\frac{\pi}{4}} \]
+\subsection{}
+\[-1 + i\sqrt{3} = \polar{2}{\frac{-\pi}{3}}\]
+\[-1 - i\sqrt{3} = \polar{2}{\frac{+\pi}{3}}\]
+\subsection{}
+\[\]
+\section{}
+\subsection{}
+\[i\]
+\subsection{}
+\[\sqrt{2} + \sqrt{2}i\]
+\subsection{}
+
+\end{document}

MA1006/tut-2023-10-17.tex

@@ -0,0 +1,9 @@
+\input{decls.tex}
+\title{Tutorial}
+\date{2023-10-17}
+\begin{document}
+\maketitle
+\section{}
+\subsection{}
+\begin{align*}
+\end{document}