CSET Subtest 1
Compound Interest
interest earned on both the principal amount and any interest already earned A = Pe^rt P is principal r is interest rate t is time Same formula for exponential growth or decay
Imaginary Numbers
Denoted by lower case i, where i is the square root of negative 1
Linearly Independent
If S = {v1 v2 v3 ... vn} is the set of vectors in Rn, then the equation k1v1 + k2v2 + k3v3 + ... knvn = 0 has at least one solution that is k1 = k2 = k3 = kn = 0 which is the trivial solution. If 0 is the only solution, then S is linearly independent A set S with two or more vectors is... Linearly Dependent: iff one of the vectors is expressible as a linear combination of the other vectors Linearly Independent: iff no vector is expressible in linear combination of the others
Whole Numbers
The natural numbers including 0 0, 1, 2, 3, 4, ...
Symmetric Property
if a=b, then b=a
Combining Functions
(f + g)(x) = f(x) + g(x) (f - g)(x) = f(x) - g(x) (fg)(x) = f(x)g(x) (f/g)(x) = f(x)/g(x) (f o g)(x) = f(g(x))
Determinant Properties
- If A has a row or column of zeroes then det(A) = 0 - det(A) = det(A^T) where A^T is the transpose of A det (kA) = k det(A) - If B is the result of interchanging two rows/columns of A, then det(B) = - det(A) - If B is the result of adding a multiple of one row of A to another, then det(B) = det(A) - If A has two proportional rows/columns, then det(A) = 0 det(AB) = det(A) det(B) det(A^-1) = 1/det(A)
Complex Conjugate
A complex number with the sign of the imaginary portion switched. e.g. a+bi and a-bi are complex conjugates
Vector Linearity Properties
A finite set that contains the 0 vector is linearly dependent A set with one vector is linearly independent A set with two vectors is linearly independent iff neither vector is a scalar multiple of the other Let S be the set with r vectors in Rn. If r > n, then S is linearly independent
Rational Function
A function in the form... f(x) = p(x) / q(x) where p(x) and q(x) are polynomials 1. May have asymptotes 2. Has a horizontal asymptote at y = 0 if n < m y = an / bm if n = m none if n > m
Abelian Group
A group that includes the commutative property
Reduced Row Echelon Form
A matrix is in reduced row echelon form when... 1. If a row is not all zeroes, then the first number in the row is a 1 (leading one) 2. Any rows consisting entirely of zeroes is at the bottom of the matrix 3. In any consecutive rows that are not all zeroes, the leading 1 in the lower row is farther to the right than the higher row 4. Each column that contains a leading 1 has zeroes everywhere else A matrix satisfying 1, 2, and 3 but not 4 is in row-echelon form.
Pascal's Triangle
A pattern for finding the coefficients of the terms of a binomial expansion. To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern. Each number is just the two numbers above it added together (except for the edges, which are all "1").
Total Ordering
A set is totally ordered under a binary relation, R, if.... 1. Reflexivity: for all elements a in the set X, aRa 2. Totality: If elements a and b are in the set X, then aRb or bRa, not both 3. Antisymmetry: If a and b are elements in the set X, such that aRb and bRa then a = b 4. Transitivity: If a,b, and c are elements in the set X, and aRb and bRc then aRc R is typically a phrase such as "is a divisor of..." or "is equal to..."
Group
A set of elements that is 1. Closed (combining any two elements produces a third element still in that set) under one operation 2. Identity Element 3. Inverse Element 4. Associativity
Ring
A set of elements that is a group and is closed under 2 operations. 1. Closed under addition and multiplication 2. Associativity of addition and multiplication 3. Commutativity of addition 4. Identity Elements 0 and 1 5. Inverse Element for addition 6. Distributivity of addition and multiplication Key Difference: No Multiplicative Inverse, No Commutative Multiplication Examples: integers, square matrices, polynomials with coefficients in a ring
Field
A set of elements that is closed under addition and multiplication 1. Associativity of addition and multiplication 2. Commutativity of addition and multiplication 3. Identity Elements 0 and 1 4. Inverse Elements -a and a^-1 5. Distributivity Examples: rational, real, and complex numbers
Systems of Linear Equations
A set of linear equations in the form a1x1 + a2x2 + a3x3 + ... anxn = b Every system has either no solutions, exactly one solution, or infinitely many solutions
Homogeneous System
A system of equations is homogeneous if all the constants b1, b2, ... bn are zero. Every homogeneous system has the trivial solution Only one of the following can be true... -system has only the trivial solution -system has infinitely many non-trivial solutions including the trivial solution A homogeneous system with more unknowns than equations has infinitely many solutions
Mathematical Induction
A two-part proof that shows a statement is true for all positive integers by 1. Showing the statement is true for n = 1 2. Assuming the statement is true for some positive integer, k, and showing it must then be true for k + 1
Inverse Property
An element in a set, such that when added or multiplied with another element in the set, results in the identity element. Usually represented by the element with the opposite sign or negative exponent. a+(-a)=0 or a*a^-1=1
Identity Property
An element in set that when added or multiplied with another element, results in that same element. 0 and 1 are identity elements for addition and multiplication a+0=a or a*1=a
Prime Factorization
Any non-prime number can be written as a product of prime numbers.
Rational Numbers
Any number that can be represented as a fraction or decimal. If a decimal, then it must either terminate or repeat. Denoted by the letter Q 4.32, 3/4, 1.3333....
Holes
Any terms that cancel out after factoring both the numerator and denominator of a rational function will have holes at those locations
Associative Property
Changing the grouping of numbers will NOT change the value. (a+b)+c=a+(b+c)
Matrix Determinants
Denoted det(A) or |A| Only works for square matrices
Fundamental Theorem of Algebra
Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers.
Vertical Asymptote
Exists at any value that makes the denominator of a simplified rational function equal zero. The function is undefined at this value because division by 0 is not allowed.
Matrix Multiplication
For two matrices, the number of columns in the first must equal the number of rows in the second. It is NOT commutative, AB does equal BA It is Associative, A(BC) = (AB)C It is Distributive, A(B + C) = AB + AC Given two matrices with dimensions n1 x m1 and m1 x n2, then multiplying the matrices will result in a n1 x n2 matrix
Augmented Matrix
Given a system of m equations in n unknowns, we can write a rectangular array that corresponds to... a11x1 + a12x2 + ... + a1nxn = b1 a21x1 + a22x2 + ... + a2nxn = b2 am1x1 + am2x2 + ... + amnxn = bn To solve, use elementary row operations - multiply a row by a non-zero constant - Interchange two rows - Add a multiple of one row to another
Vector Norm/Magnitude
Given a vector, v = <v1, v2, v3> ||v|| = (v1^2 + v2^2 + v3^2)^(1/2) ||kv|| = |k|*||v|| v/||v|| is the vector of length one in the direction of v
Vector Properties
Given u, v, w in R2 or R3 and scalars k and j, then... u + v = v + u (u + v) + w = u + (v + w) u + 0 = u u + -u = 0 k(ju) = (jk)u k(u + v) = ku + kv (k +j)u = ku + ju
Horizontal Asymptote
Horizontal line that describes the "end behavior" of a rational function. Occurs if the degree of the numerator of a rational function less than or equal to the degree in the denominator
Cramer's Rule
If AX = B is system of n linear equations in n unknowns such that det(A) does not equal 0, then the system has a unique solution which is... x1 = det(A1)/det(A) x2 = det(A2)/det(A) xn = det(An)/det(A) Where Aj is the matrix obtained by replacing the jth column of A with the column matrix B.
Rational Root Theorem
If P(x) is a polynomial with integer coefficients, and if p/q is a rational root of P(x) where p and q are relatively prime, then p is a divisor of a sub 0, and q is a divisor of a sub n.
Complex Conjugates Theorem
If P(x) is a polynomial with real coefficients, and a+bi is an imaginary root of the equation P(x)=0, then a-bi is also a root.
Remainder Theorem
If a polynomial P(x) is divided by (x-c), then the remainder is P(c) Proof: P(x) = (x-c)*Q(x) + R where Q(x) is the quotient of P(x) divided by (x-c). Therefore... P(c) = (c-c)*Q(c) + R P(c) = 0*Q(c) + R P(c) = R
Transitive Property
If a=b and b=c, then a=c
Addition Property
If a=b, then a+c=b+c
Multiplication Property
If a=b, then ac=bc
Factoring Trinomials
If in the form x^2 + bx + c, then it can be factored into (x + m)(x + n) where c = mn and b = m + n
Dot Product
If u and v are vectors in R2 or R3, then the dot product is... u . v = ||u||*||v||*cos (theta) where theta is the angle between u and v, and 0<theta<pi 1. Result is a scalar quantity, not a vector 2. If u . v = 0 then u and v are orthogonal, or theta = pi/2 3. If u and v are non-zero, then theta is... acute if u . v is greater than 0 obtuse if u . v is less than 0 90 if u . v is equal to 0
Cross Product
If u and v are vectors in R3 such that u = <u1,u2,u3> and v = <v1,v2,v3> then the cross product of u and v is... u x v = <u2v3-u3v2, u3v1-u1v3, u1v2-u2v1> ||u x v|| = ||u|| ||v|| sin (theta) ||u x v|| is equal to the area of the parallelogram formed with u and v For unit vectors i x j = k, j x k = i, k x i = j
Parallelogram Rule
If v an u are vectors and are positioned so that their initial points coincide, then they form adjacent sides of a parallelogram u + v = v + u
Integers
Includes positive and negative natural numbers. Denoted by the letter Z -2, -1, 0, 1, 2, ...
Inverse Functions
Let f be a one to one function with domain Df and range Rf. A function g is the inverse of f if Dg = Rf and Rg = Df 1. y = f(x) iff x = g(y) 2. g = f^-1 3. (f o g)(x) = x and (g o f)(x) = x 4. f^-1 is the reflection of f about the line y = x 5. A function is even, if f(-x) = f(x) 6. A function is odd, if f(-x) = -f(x)
Vector Components
Let v be vector in R3 with initial point at (0,0,0) at terminal point (v1, v2, v3). The coordinates of the terminal point, (v1, v2, v3) are called the components of v v = < v1, v2, v3> Given v = < v1, v2, v3> and u = <u1, u2, u3>, then v = u if v1 = u1, v2 = u2, v3 = u3 v + u = <v1+u1, v2+u2, v3+u3> kv = <kv1, kv2, kv3>
Matrix Addition
Matrices must be the same dimensions
Scalar product
Multiplication of a vector and a scalar, k
Complex Number Properties
Multiplying a + bi times a - bi results in a^2 + b^2 which is the magnitude of z squared, or the modulus
Natural Numbers
No Decimals or Fractions, must be greater than 0 1, 2, 3, 4, ...
Complex Numbers
Numbers containing both a real and imaginary part. Written in the form a+bi, where a is real and bi is imaginary
Irrational Numbers
Numbers that cannot be expressed as fraction. A decimal number that never ends and does not repeat. Denoted by the letter I pi
Inverse Matrix
Only calculated for square matrices A matrix is not invertible of det(A) = 0 A x A^-1 = I Create an augmented matrix with the identity matrix on the right and then use Gauss-Jordan elimination to turn the left side into the identity matrix which will leave the right side as A^-1
Real Numbers
Set of all rational, irrational, integer, whole, and natural numbers. Denoted by the letter R
Unit Vectors
Special vectors that have a magnitude of 1. In R2, these include horizontal basis vector, i, and vertical basis vector, j. i = <1, 0> j = <0, 1> In R3, i = <1, 0, 0> j = <0, 1, 0> k = <0, 0, 1> all vectors u = <a, b, c> can be written as... u = ai + bj + ck
Identity Matrix
The identity matrix, I, is the matrix with 1's on the diagonal and zeroes everywhere else. Functions like 1 for multiplication A x I = A
Divisor
The number by which another number is divided.
Gauss-Jordan Elimination
The procedure used to change the augmented matrix into RREF is as follows: 1. Locate the leftmost column that does not consist entirely of zeroes 2. Interchange the top row with another row, if necessary, to bring a non-zero entry to the top of the column from step 1 3. If the entry at the top of the column is a, then multiply the first row by 1/a to produce a leading 1 4. Add suitable multiples of the top row to the rows below so that all entries below the leading 1 become zeroes 5. Repeat steps 1-4 for the rest of the rows until the matrix is in row-echelon form.
Commutative Property
The property that says that two or more numbers can be added or multiplied in any order without changing the result. a+b=b+a and a*b=b*a
Matrix Trace
The trace of a square matrix, tr(A) is the sum of the entries on the main diagonal
Matrix Transpose
The transpose of an m x n matrix, A, is a matrix A^-1 of n x m dimensions
Euclidean Algorithm
Used to find the greatest common divisor of a and b. Iterative process that a = q0b + r0 b = q1r0 + r1 r0 = q2r1 + r2 Repeat until r sub n equals 0. The greatest common divisor will then be r sub n-1
Modular Addition/Multiplication
a + b(mod n) = r 1. n is the number of elements in the set, including 0 2. r is the remainder when a + b or a*b is divided by n 3. Only applied if a + b or a*b results in a number outside the given set 4. If is NOT prime, its divisors will have no multiplicative inverse
Reflexive Property
a = a
Binomial Theorem
a formula for finding any power of a binomial without multiplying at length. Combinatorial Notation: (n k) = n! / k!(n-k)! where n is greater than or equal to k n! / k! = n(n-1)(n-2)....(k+1) (x+y)^n = the sum from k = 0 to n of (n k)*x^(n-k)*y^k Which results in... x^n + nx^n-1*y +
One-to-One Function
a function where each element of the range is paired with exactly one element of the domain If no horizontal line can be drawn that intersects the graph in more than one point then the function is one to one.
Synthetic Division
a method used to divide a polynomial by a binomial
Polynomial Long Division
a method used to divide polynomials similar to the way you divide numbers
Distributive Property
a property in which multiplication is applied to addition of two or more numbers where each term inside a set of parentheses can be multiplied by a factor outside the parentheses a*(b+c)=ab+ac
Descartes' Rule of Signs
a technique for finding the number of positive and negative roots to a polynomial equation with real coefficients
Completing the Square
a way to convert a quadratic equation in standard form into perfect square form
Difference of Squares
a^2 -b^2 = (a+b)(a-b)
Discriminant
b²-4ac If b²-4ac is... zero: only 1 real root with vertex on the x-axis positive: 2 real roots with 2 x-intercepts negative: 2 complex roots with no x-intercepts
Vectors
quantities that have both a magnitude and a direction equivalent vectors have the same direction and magnitude
Exponent Properties
see photo
Logarithm Properties
see photo
Vector Addition
the sum of two vectors A and B is a vector C which when the tail of B is connected to the head of A, will have the same tail as A and the same head as B A + B = C
Dot Product Properties
u . v = v . u u . (v + w) = u . v + u . w k(u . v) = (ku) . v = u . (kv) v . v > 0 if v does not equal 0 v . v = 0 if v = 0 u . v = 0 iff u and v are orthogonal
Cross Product Properties
u x v = v x u u x (v + w) = (u x v) + (u x w) k(u x v) = (ku) x v = u x (kv) u x 0 = 0 u x u = 0 u x v = 0 iff u and v are parallel
Vertex Form
y=a(x-h)^2+k Vertex is at (h, k) In standard form, the vertex will be at (-b/2a, f(-b/2a))
