(incomplete)
| Part 1: Matrices | a matrix is a rectangular array of numbers forming m rows and n columns enclosed in [ brackets ]
denoted by capital letters each no. is called an element size/order/dimension is given by m x n |
| Special Matrices | Row matrix - formed by only one row. m = 1 Column matrix - formed by only one column. n = 1 Square matrix - m = n |
| Transpose of a Matrix | Interchange the rows and columns, becomes a n x m order. Write the first row as the first column, and so on. |
| Operations | Adding - add corresponding elements. can only be done with matrices w/ the same order and dimension
Subtracting - same as adding, just subtract corresponding elements (use the position of element) Multiplying - if scalar, (eg. 2B), just multiply every element in B by 2. *if non-scalar, (eg. get AB), non-commutative. n of A must be equal to m of B. In other words, no. of columns ng first matrix dapat pantay sa no. of rows ng second matrix |
| Non-scalar multiplication |   angkulit, nagmukhang twister na game... ..ah ulul |
| Determinant | a special no. associated with a square matrix. something to do with the value of all the numbers in
the matrix di ko matatapos tong part na to, hirap eh |
| Part 2: Functions I | intro, domain and range, function and relation |
| Domain and Range | sa lessons natin dati pa, pag binigyan ka ng group of ordered pairs (coordinates), domain
refers to all the first coordinates of the pairs. abscissa, x. basta't ung first number sa pair, ung
nakaplot sa x-axis. all first coordinates are part of the domain. on the other hand, range refers to the set of all second numbers in the ordered pairs. pde rin ordinate, y. lahat yan basta't second number sa pair ay part ng range. |
| Functions and Relations | In essence, all equations are relations. A function is a special kind of relation wherein
(in simple terms), a vertical line passes through the graph only once. this has several implications
on the domain and range of the relation: in a relation, there are no rules. kahit anong combination ng domain and range pde, miske ano shape kalabasan ng graph ayus lang. anything na may two variables affecting each other ay relation, kac ung x and y ay merong relation. yun lang however, sa function, since kelangan pumasa sya sa vertical line test, meron restrictions: cannot have a repeating value in the domain. dapat sa lahat ng x, walang uulit. repeating y values are allowed 1-1 basis allowed (for each unique x-value, may unique y-value, walang ulit kahit ano) allowed graphs: lines (except vertical), parabola (up-down concavity only) disallowed graphs: all other conics, vertical lines in functions, the y variable cannot have an even exponent. |
| function | when you say f(x) = some statements, it means that the statements in the right side of the equation
are a function of x. the statements can be any combination of terms which follow the rules stated in
the preceeding section, and have the variable of x in at least one term. and when you give a value to
x, you substitute that value for x in the equation. for example: f(x) = 3x + 2 if you want to find f(7) you simply substitute 7 for x magiging: f(7) = 3(7) + 2 f(7) = 23 |
| Domain and Range restrictions | for finding the domain, write y in terms of x. what values of x will still make y true? y = [some x stuff] ang only restrictions dyan is if [ y = something where x is inside a radical sign ] or if [ y = something where x is in the denominator ]. in that case, pag nasa radical sign ung x, the whole qty inside the radical should not be negative kya kung y = √(something) edi (something) should be greater than or equal to zero or (something) ≥ 0 pag nasa denominator naman ung buong denominator di pde maging zero so the whole denominator should not be equal to zero so not-equate it to zero or (denominator) ≠ 0 for range naman, first write x in terms of y. transpose to get x on one side only. follow the same rules as for the domain pero obviously y na ngayon ung napapaloob sa radical or nasa denominator |
| Part 3: Functions II | operations, odd even, inverse functions |
| Odd and Even | basically to tell the difference between odd and even functions you substitute the negative of the
variable in the parentheses in the equation. kung ganun parin ung lumabas then the function is even.
usually it follows na if sa lahat ng terms ung variable ay may even exponent, even function din. kung odd function naman, when you substitute the negative of the variable for that variable in the function, tapos ang lumabas ay exactly the same function but with opposite signs in all terms edi odd function sya. pde rin neither even nor odd, di cya exactong un function na un, di rin cya exact opposite, pareho sign for some terms, opposite for others |
| Operations on Functions | For addition and subtraction, just add the terms together. (pag subtraction change all signs of the
second function before adding). pag multiplication distribute all terms of the first function to all terms of second function pag division just write the first term over the 2nd, try to cancel out stuff by factoring sa composition, first get the value of the innermost function then use that value for the next innermost function, moving outward hanggang sa maresolve lahat ng functions |
| Inverse Function | begin with y = f(x) basically get the y to one side of the equation, or kung f(x) = [something] edi
pde mo na isulat y = [ung something na un] interchange all x with y. lahat ng x gawin mo y tapos ung y sa left side gawin mo x solve for y (ung naka-interchange na, ung x = [something]) ung makukuha mo na end result is equal to f-1(x) |