CSET Subtest 1

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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))


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