diff --git a/CS1029/notes-2023-11-01 b/CS1029/notes-2023-11-01 new file mode 100644 index 0000000..032579f --- /dev/null +++ b/CS1029/notes-2023-11-01 @@ -0,0 +1,6 @@ +Edge contractions: merge two vertices, removing an edge between them +Representations: adjacency list + adjacency matrix + What does the determinant of an adjacency matrix mean? + incidence matrix: vertices against edges, 1 where edge is connected to vertex + diff --git a/CS1029/notes-2023-11-07 b/CS1029/notes-2023-11-07 new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/CS1029/notes-2023-11-07 @@ -0,0 +1 @@ + diff --git a/CS1029/notes-2023-11-08 b/CS1029/notes-2023-11-08 new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/CS1029/notes-2023-11-08 @@ -0,0 +1 @@ + diff --git a/CS1029/notes-2023-11-14 b/CS1029/notes-2023-11-14 new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/CS1029/notes-2023-11-14 @@ -0,0 +1 @@ + diff --git a/CS1029/practical-2023-11-03/relations b/CS1029/practical-2023-11-03/relations new file mode 100644 index 0000000..180da4b --- /dev/null +++ b/CS1029/practical-2023-11-03/relations @@ -0,0 +1,58 @@ +1. a) {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (2, 3), (2, 4), (2, 5), (3, 3), (3, 4), (3, 5), (4, 4), (4, 5), (5, 5) } + Reflexive: \forall x. (x, x) \in R + Antisymmetric (& not symmetric): all pairs have x <= y. forall x, y. x <= y, !(y <= x) || y == x + Transitive + b) {(1, 2), (1, 4), (2, 1), (2, 3), (3, 2), (3, 4), (4, 1), (4, 3)} + Not reflexive: there are no pairs of (x, x) + Symmetric - all pairs have their reverse represented + Not antisymmetric: symmetric and anti-symmetric are mutually exclusive + Not transitive: (1, 2) and (2, 3) - 1 + 3 is not odd + +2. {(item, quantity)} + {(Name, {(key, value)})} + {(Name, Address, {(Room type, price, {(key, value)})}, ...)} + +3. 105 305 306 505 705 707 905 906 909 + +4. a) ab ac bc cb + b) {(a, a), (a, b), (a, c), (b, b), (b, a), (b, c), (c, a), (c, b), (d, d)} + +5.a) {(1, 1), (1, 2), (1, 4), (2, 1), (2, 3), (3, 2), (3, 3), (3, 4), (4, 1), (4, 3), (4, 4)} + Not reflexive: (2, 2) is not present + Symmetric, therefore not anti-symmetric + Not transitive: (1, 2) and (2, 3), but not (1, 3) + + b) 12 21 14 41 32 23 43 34 + Not reflexive + Symmetric, therefore not antisymmetric + Not transitive: 12 and 23 but not 13 + +6. 1 1 0 0 + 1 1 0 0 + 1 0 1 1 + 0 0 0 1 + +Yes, it's reflexive + +8. {(a, b) | a divides b OR b divides a} +9. No, it's not transitive. (a, b) & (b, d), but not (a, d) +10. a) Not equivalence relation: missing transitivity + (1, 3) and (3, 2), but not (1, 2) + b) {0}, {1, 2}, {3} +11. \forall n \in N_0: + 0 + 3n + 1 + 3n + 2 + 3n + +12. a) Y + b) N: 0 is in both - not disjoint + c) Y + d) N: 0 is missing + +13. a) 00 11 22 33 44 55 12 21 34 43 35 53 45 54 + b) 00 11 22 33 44 55 01 10 23 32 45 54 + c) 00 11 22 33 44 55 01 10 02 20 12 21 34 43 35 53 45 54 + +14. a) Y, trivially + b) N: not antisymmetric ((2, 3) and (3, 2)) + c) N: not reflexive (no (3, 3)) diff --git a/CS1029/practical-2023-11-10/graphs b/CS1029/practical-2023-11-10/graphs new file mode 100644 index 0000000..c077e7b --- /dev/null +++ b/CS1029/practical-2023-11-10/graphs @@ -0,0 +1,64 @@ +1. Undirected, unlooped, multi-edged: multigraph +b) directed, looped, multi-edged: directed pseudo-multigraph + +2. ac bd +b) cd cd dd ee ab bc + +3. {{paper}} + +4. vertices: 6 + edges: 6 + degree: a: 2 b: 4 c: 1 f: 3 e: 2 d: 0 + isolated: d + pendant: c +b) vertices: 5 + edges: 14 + degree: a: 6 b: 6 c: 6 d: 5 e: 3 + isolated: - + pendant: - + +5. vertices: 4 + in-a : 2 + out-a: 2 + in-b: 3 + out-b: 4 + in-c: 2 + out-c: 1 + in-d: 1 + out-d: 1 + +6. {ac} {bde} +b) Not bipartite: 3-loop bcf would require 3 sets + +7. {{ paper }} + +8. a -> abcd + b -> d + c -> ab + d -> bcd + +9. | a b c d + --+-------- + a | 1 1 1 1 + b | 0 0 0 1 + c | 1 1 0 0 + d | 0 1 1 1 + +10. {{ paper }} + +11. v1 -> u1 + v2 -> u4 + v3 -> u2 + v4 -> u5 + v5 -> u3 + +12. v1 -> u4 + v2 -> u3 + v3 -> u1 + v4 -> u2 + +13. PSCL + a) YNN4 + b) N--- + c) N--- + d) YYY5 diff --git a/CS1032/notes-2023-11-02 b/CS1032/notes-2023-11-02 new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/CS1032/notes-2023-11-02 @@ -0,0 +1 @@ + diff --git a/CS1032/notes-2023-11-03 b/CS1032/notes-2023-11-03 new file mode 100644 index 0000000..65e2f4e --- /dev/null +++ b/CS1032/notes-2023-11-03 @@ -0,0 +1 @@ +Pencil & eraser for final exam \ No newline at end of file diff --git a/CS1032/notes-2023-11-09 b/CS1032/notes-2023-11-09 new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/CS1032/notes-2023-11-09 @@ -0,0 +1 @@ + diff --git a/CS1032/practical-2023-11-01/head.py b/CS1032/practical-2023-11-01/head.py new file mode 100644 index 0000000..f3c2bdf --- /dev/null +++ b/CS1032/practical-2023-11-01/head.py @@ -0,0 +1,21 @@ +import sys, itertools + +def open_arg(_arg0, filename = "-"): # Default of - means stdin + if filename == "-": + return sys.stdin + + return open(filename, 'r') + +def head(file): + for line in file.readlines()[:10]: + print(line, end = "") + +if __name__ == '__main__': + try: + file = open_arg(*sys.argv) + head(file) + except FileNotFoundError: + print("File not found!", file=sys.stderr) + sys.exit(1) + except: + print("Some other error occured", file=sys.stderr) \ No newline at end of file diff --git a/CS1032/practical-2023-11-01/tail.py b/CS1032/practical-2023-11-01/tail.py new file mode 100644 index 0000000..8eaa059 --- /dev/null +++ b/CS1032/practical-2023-11-01/tail.py @@ -0,0 +1,21 @@ +import sys, itertools + +def open_arg(_arg0, filename = "-"): # Default of - means stdin + if filename == "-": + return sys.stdin + + return open(filename, 'r') + +def tail(file): + for line in file.readlines()[-10:]: + print(line, end = "") + +if __name__ == '__main__': + try: + file = open_arg(*sys.argv) + tail(file) + except FileNotFoundError: + print("File not found!", file=sys.stderr) + sys.exit(1) + except: + print("Some other error occured", file=sys.stderr) \ No newline at end of file diff --git a/CS1032/practical-2023-11-08/calc.py b/CS1032/practical-2023-11-08/calc.py new file mode 100644 index 0000000..f8cf840 --- /dev/null +++ b/CS1032/practical-2023-11-08/calc.py @@ -0,0 +1,70 @@ +# This function adds two numbers +def add(x, y): + return x + y + +# This function subtracts two numbers +def subtract(x, y): + return x - y + +# This function multiplies two numbers +def multiply(x, y): + return x * y + +# This function divides two numbers +def divide(x, y): + return x / y + +def avg(x, y): + return (x + y) / 2 + +def sci(x, y): + return x * 10 ** y + + +print("Select operation.") +print("1.Add") +print("2.Subtract") +print("3.Multiply") +print("4.Divide") +print("5.Average") +print("6.Scientific Notation") + +while True: + # take input from the user + choice = input("Enter choice(1/2/3/4/5/6): ") + + # check if choice is one of the four options + if choice in ('1', '2', '3', '4', '5', '6'): + try: + num1 = float(input("Enter first number: ")) + num2 = float(input("Enter second number: ")) + except ValueError: + print("Invalid input. Please enter a number.") + continue + + if choice == '1': + print(num1, "+", num2, "=", add(num1, num2)) + + elif choice == '2': + print(num1, "-", num2, "=", subtract(num1, num2)) + + elif choice == '3': + print(num1, "*", num2, "=", multiply(num1, num2)) + + elif choice == '4': + print(num1, "/", num2, "=", divide(num1, num2)) + + elif choice == '5': + print(f"avg({num1}, {num2})", "=", avg(num1, num2)) + + elif choice == '6': + print(f"{num1}e{num2}", "=", sci(num1, num2)) + + # check if user wants another calculation + # break the while loop if answer is no + next_calculation = input("Let's do next calculation? (yes/no): ") + if next_calculation.lower().startswith('n'): + break + else: + print("Invalid Input") + diff --git a/MA1006/decls.tex b/MA1006/decls.tex index 89327a8..9a2f722 100644 --- a/MA1006/decls.tex +++ b/MA1006/decls.tex @@ -8,8 +8,9 @@ \date{} \author{} -\renewcommand{\Re}[1]{\operatorname{\mathbb{R}e}(#1)} -\renewcommand{\Im}[1]{\operatorname{\mathbb{{I}}m}(#1)} +\newcommand{\paren}[1]{\left(#1\right)} +\renewcommand{\Re}[1]{\operatorname{\mathbb{R}e}\paren{#1}} +\renewcommand{\Im}[1]{\operatorname{\mathbb{{I}}m}\paren{#1}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} @@ -18,8 +19,8 @@ \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}}}} +\newcommand{\adj}[1]{\operatorname{adj}#1} \makeatletter \renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{% diff --git a/MA1006/linear.tex b/MA1006/linear.tex index cc9a7c7..7b860ff 100644 --- a/MA1006/linear.tex +++ b/MA1006/linear.tex @@ -5,239 +5,354 @@ \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 + \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 -}\\\\ + & \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} -}\\\\ + & \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} -}\\\\ + & \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} -}\\\\ + & \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. + \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 + \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} -} + & \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. + & \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 \[A = \begin{pmatrix} - a_{11} & a_{12} & a_{1n} \\ - a_{21} & \ddots & \vdots \\ - a_{31} & \ldots & a_{3n} \\ -\end{pmatrix} + a_{11} & a_{12} & a_{1n} \\ + a_{21} & \ddots & \vdots \\ + a_{31} & \ldots & a_{3n} \\ + \end{pmatrix} \] \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 $A_{ij}$ + \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} + & \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 - $d_{ij} = 0 \quad \forall{i \neq j}$ \\ -$\det{A} = \prod^{N}_{i=0}~a_{ij} \forall~\text{ row-echelon }A$ + \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 + \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[$A^T$] $a^T_{ij} = a_{ji}~ \forall~i,j$ + \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 +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 \\ + 2 & 1 + 1 & 3 + 6 \\ 4(2) & 4 + 1 & 3(4) + 6 \\ - 0 & 2 & 2(6) \\ + 0 & 2 & 2(6) \\ \end{pmatrix} = \begin{pmatrix} - 2 & 2 & 9 \\ - 8 & 5 & 18 \\ - 0 & 2 & 12 - \end{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} + \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*} + +\[A\vec{x} = \vec{b}\] +where $A$ is the coefficient matrix, $\vec{x}$ is the variables, and $\vec{b}$ is the values of the equations of a linear equation system. +\subsection*{Inverse Matrices} +The identity matrix exists as $I_n$ for size $n$. +\[AA^{-1} = I_n = A^{-1}A \quad \forall~\text{matrices }A \text{ of size } n\] +Assume that $A$ has two distinct inverses, $B$ and $C$. +\begin{align*} + & \text{matrix multiplication is associative} \\ + \therefore~ & C(AB) = (CA)B \\ + \therefore~ & C I_n = I_n B \\ + \therefore~ & C = B \\ + & \text{ + As $B = C$, while $B$ and $C$ are assumed to be distinct, matrices have no more than one unique inverse by contradiction + } +\end{align*} +Matrices are invertible $\iff \det{A} \neq 0$ +\[\det{AB} = \det{A}\det{B}\] +\[\therefore~ \det{A}\det{A^{-1}} = \det{I_n} = 1\] +\[\therefore~ \det{A} \neq 0 \] +\begin{align*} + \begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix} +\end{align*} +\subsubsection*{Computation thereof} +\[\det{A} = \sum_{k = 1}^{n}~a_{ik}(-1)^{i+j}\det{A_{ij}} \quad \text{ for any $i$}\] +\begin{description} + \item[Matrix of Cofactors: $C$] determinants of minors \& signs of laplace expansion \\ + ie. $\sum A \odot C = \det{A}$ + \item[$\adj{A}$ Adjucate of $A$ =] $C^T$ +\end{description} +\begin{align*} + A & = \begin{pmatrix} + 1 & 0 & 1 \\ + -1 & 1 & 2 \\ + 2 & 0 & 1 + \end{pmatrix} \\ + C(A) & = \begin{pmatrix} + 1 & 5 & -2 \\ + 0 & -1 & 0 \\ + -1 & -3 & 1 \\ + \end{pmatrix} +\end{align*} +$$ A^{-1} = \frac{\adj{A}}{\det{A}} $$ +Gaussian elimination can also be used: augmented matrix with $I_n$ on the right, +reduce to reduced row-echelon. If the left is of the form $I_n$, the right is +the inverse. If there is a zero row, $\det{A} = 0$, and the $A$ has no inverse. +\section*{Linear Transformations} +\begin{align*} + f: & ~ \R^n \to \R^m \\ + f & (x_1, \cdots, x_n) = (f_1(x_1, \cdots, x_n), f_2(x_1, \cdots, x_n), \cdots, f_m(x_1, \cdots, x_n)) +\end{align*} +$f$ is a linear transformation if \(\forall i.~f_i(x_1, \cdots, x_n)\) is a +linear polynomial in $x_1, \cdots, x_n$ with a zero constant term +\begin{align*} + f(x_1,~ x_2) & = (x_1 + x_2,~ 3x_1 - x_2,~ 10x_2) \tag{is a linear transformation} \\ + g(x_1,~ x_2,~ x_3) & = (x_1 x_2,~ x_3^2) \tag{not a linear transformation} \\ + h(x_1,~ x_2) & = (3x_1 + 4,~ 2x_2 - 4) \tag{not a linear transformation} \\ +\end{align*} +\[f: \R^n \to \R^m = \vec{x} \to A\vec{x} \] +\[\exists \text{ a matrix $A$ of dimension $n$x$m$ } \forall\text{ linear transforms } f \] +\[\forall \text{ matrices $A$ of dimension $n$x$m$ } \exists \text{ a linear transform $f$ of dimension $n$x$m$ such that } f(\vec{x}) = A\vec{x} \] +Function composition of linear translations is is just matrix multiplication: +\begin{align*} + f(\vec{x}) & = A\vec{x} \\ + g(\vec{y}) & = B\vec{y} \\ + (f\cdot g)(\vec{x}) & = g(f(\vec{x})) = BA\vec{x} +\end{align*} +A function \(f: \R^n \to \R^m\) is a linear transformation iff: +\begin{enumerate} + \item $f(\vec{x} + \vec{y}) = f(\vec{x}) + f(\vec{y}) \quad \forall~\vec{x},~\vec{y} \in \R^n $ + \item $f(r\vec{x}) = r\cdot f(\vec{x}) \quad \forall~\vec{x} \in \R^n, r \in \R $ +\end{enumerate} +\subsection*{Building the matrix of a linear transform} +\[ f(\vec{x}) = f(x_1\vec{e}_1 + x_2\vec{e}_2) = f(x_1\vec{e}_1) + f(x_2\vec{e}_2) = x_1f(\vec{e}_1) + x_2f(\vec{e}_2) \] +\[ A = \begin{pmatrix} f(\vec{e}_1) & f(\vec{e}_2) \end{pmatrix} \] +\begin{align*} + & \vec{e}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \\ & \vec{e}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix} + \\ & \vdots + \\ & \forall \vec{x}.~ \vec{x} = \sum_{i}^{n}~\vec{e}_i x_i +\end{align*} +\subsection*{Composition} +\[ \paren{f \cdot g}\paren{\vec{x}} = f(g(\vec{x})) = AB\vec{x} \] +where: $f(\vec{x}) = A\vec{x}$, $g(\vec{x}) = B\vec{x}$ +\subsection*{Geometry} +\begin{description} + \item[rotation of $x$ by $\theta$ anticlockwise] \( = R_\theta = \begin{pmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{pmatrix} \) + \item[reflection about a line at angle $\alpha$ from the $x$-axis] \( = T_\alpha = R_{\alpha}T_0R_{-\alpha}\) where \( T_0 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \) + \item[scaling by $\lambda \in \R$] \( = S_\lambda = \lambda I_n\) + \item[Skew by $\alpha$ in $x$ and $\gamma$ in $y$] \( \begin{pmatrix} \alpha & 0 \\ 0 & \gamma \end{pmatrix}\) +\end{description} +The image of the unit square under the linear transform $A$ is a parallelogram of $(0, 0)$, $(a_{11}, a_{21})$, $(a_{12}, a_{22})$, $(a_{11} + a_{12}, a_{21} + a_{22})$, with area $ \abs{\det{A}} $ +\subsection*{Inversion} +Inversion of a linear transformation is equivalent to inversion of its representative matrix +\subsection*{Eigen\{values, vectors\}} +\[ \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} a \\ 0 \end{pmatrix} = a\vec{e}_1\] +\[ \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}\begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ b \end{pmatrix} = b\vec{e}_2\] +\[ T_\alpha \vec{x} = \vec{x} \text{ for $\vec{x}$ along the line of transformation }\] +\begin{description} + \item[Eigenvector (of some transformation $f$)] A non-zero vector $\vec{x}$ such that $f(\vec{x}) = \lambda\vec{x}$ for some value $\lambda$ + \item[Eigenvalue] $\lambda$ as above +\end{description} +\[ \forall \text{ eigenvectors of $A$ } \vec{x}, c \in R, \neq 0 .~ c\vec{x} \text{ is an eigenvector with eigenvalue } \lambda\] + +\[ \forall A: \text{$n$x$n$ matrix}.\quad P_A\paren{\lambda} = \det{\paren{A - \lambda I_n}} \tag{characteristic polynomial in $\lambda$}\] +Eigenvalues of $A$ are the solutions of $P_A\paren{\lambda} = 0$ +\begin{align*} + & A\vec{x} = \lambda\vec{x} & x \neq 0\\ + \iff & A\vec{x} - \lambda\vec{x} = 0 \\ + \iff & (A - \lambda I_n)\vec{x} = 0 \\ + \iff & \det{\paren{A - \lambda I_n}} = 0 \\ + & \quad \text{ or $\paren{A - \lambda I_n}$ is invertible and $x = 0$ } +\end{align*} +\[ P_{R\theta}(\lambda) = \frac{2\cos{\theta} \pm \sqrt{-4\lambda^2\sin^2{\theta}}}{2}\] +\[ R_\theta \text{ has eigenvalues }\iff \sin{\theta} = 0 \] \end{document} \ No newline at end of file