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74 Cards in this Set
- Front
- Back
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What is a field?
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a set F with two operations (+ & x )
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List properties of Field
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F is closed under + & x ,
+ & x are associative, + & x are communative + & x have identity elements 0 & 1 in F every element has an additive inverse in F every non-zero element has multiplicative inverse in F x distributes over + |
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F is closed under + & x
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Means that a + b and ab are elements of F
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+ & x are associative
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(a+b) + c = a + (b+c) & (ab)c = a(bc)
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+ and x are commutative
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a+b = b+ a and ab=ba
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+ and x have identity elements 0 &1 in F respectively
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a+0 = a and b*1 = b
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every element has an additive inverse in F
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For every a in F, there is an element -a in F satisfying a+(-a) = 0
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every non-zero element has a multiplicative inverse in F
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If b does not equal 0, then there is 1/b in F satisfying b(1/b) =1
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x distributes over +
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a(b+c) = ab + ac
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What are some most common fields
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Rational numbers, real numbers, complex numbers
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What is a ring
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A set R together with two operations (_ & x) satisfying the following properties
1. R is closed under _ and x 2. + and x are associative 3. + is commutative 4. + and x have identity elements 0 and 1 in R, respectively 5. every element has an additive inverse in R 6. x distributes over + |
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Two key differences between the definition of a ring and field
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1. Rings do not necessarily have a commutative multiplication
2. rings do not necessarily contain multiplicative inverses * every field is a ring, but not all rings are fields |
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Name some common rings
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intergers, set of square matrices (2x2) and set of polynomials with coefficients in another ring
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Some non-examples of fields and rings
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Common rings that are not fields: intergers, set of square matrices , set of polynomials with coefficients in another ring
natural numbersr are not a ring Set of all quadratic polynomials are not rings |
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Give an example of ring that is not a field
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intergers, polynomials, set of all square matrices of a given size
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Why are integers not a field?
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because not ever integer has a multiplicative inverse that is an integer. (e.g. multiplicative inverse of 3 is 1/3, which is not an integer)
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Why are natural numbers not a ring?
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Because they do not contain an additive identity
Also, the additive inverses of natural numbers are not natural numbers |
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why is the set of all quadratic polynomials not a ring
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Set of quadratic polynomials is not closed under multiplication.
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Write multiplicative inverse of 3-2i in a+bi form
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1/(3/2i) * (3+2i) / (3+2i) = (3+2i) / (9-4i^2) = (3+2i) / 13 = 3/13 + 2i/13
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What is a non-commutative ring?
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A ring satisfies having closure under addition and multiplication ( expand this one more later)
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Prove that set of 2 x 2 invertible matrices with complex number entries is NOT a field
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Expand this one later
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Write multiplicative inverse of x _ yi in a_bi form
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expand later
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What is the smallest field that contains 0 and 1?
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Do on paper
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Using the field properties listeved above, prove that (a_ b) c - ac _ bc
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(a_b) c = c(a_b) = ca +cb = ac+bc
here they used commutativit yof multiplication and distributive property |
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Let F be the field of rational functions p(x) / q(x) , where these functions are any polynomials with real coefficients and q(x) does not equal zero
1. show that F contains a multiplicative identity element. 2. Show that F is closed under multiplication 3. Show that F is closed under addition 4. Show that every non-zero element of F is invertible |
Show on paper
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What are some basic properties of real and complex numbers?
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FIeld properties are most basic properties of real and complex numbers
Rela numbers, there are properties of ordering, like Trichotomy Axiom a. if a < b and c< d, then a+c < b+d b. If a < b and c>0, then ac < bc c. If a < b and c< 0, then ac > bc d. If a<- b and b<- a, then a = b e. If a < b and b< c, then a < c f. If a and b are real numbers, then exactly one of the following is true : a < b, a > b, or a = b |
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List Properties of equality
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Reflexive, symmetric, transitive properties of equality, additive property of equality, multiplicative property of equality
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What is the definition of a rational number... of a complex number?
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A rational number can be expressed as the ratio of two integers. Any rational number can be written as p/q, where p and q are integers, and q cant equal 0
A complex number can be expressed as sum of real number and imaginary number e.g. a_bi, where a & b are real numbers and i^2 = -1 |
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How can a complex number be expressed?
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It can be expressed as the sum of a real number and an imaginary number.
a+bi |
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What is an imaginary number?
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An imaginary number is one satisfying x^2 <- 0.
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Explain why solution 3x-5 = 4 is x=3 by showing each step
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3x-5 = 4 GIVEN
(3x - 5) + 5 = 4+5 ADDITIVE Property of Equality (3x-5) +5 = 9 Arithmetic (3x + (-5) + 5 = 9 Definition of Subtraction 3x + ((-5) +5) =9 Associative property of Addition 3x + 0 =9 Additive Inverse 3x = 9 Additive Identity 1/3 (3x) = 1/3 (9) Multiplicative Property of Equality 1/3 (3x) = 3 Arithmetic (1/3 * 3) x = 3 1x = 3 Multiplicative Inverse x = 3 Multiplicative Identity |
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Explain why the solution to -3x - 5 < 4 is x>-3 by showing each step
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-3x - 5 < 4 Given
(-3x - 5) + 5 < 4+5 Additive Propperty of Inequality (-3x - 5) + 5 < 9 Arithmetic (-3x + (-5)) + 5 < 9 Definition of Subtraction -3x + ((-5) +5) < 9 Associative Property of Addition -3x + 0 < 9 Additive Inverse -3x < 9 Additive Identity -1/3 (-3x) > -1/3(9) Multiplicative Property of Inequality -1/3(-3x) > -3 Arithmetic (-1/3 * -3) x > -3 Associative Property of Multiplication 1x > -3 Multiplicative Inverse x > -3 Multiplicative Identity |
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Pg 9 c)
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pg 10 d)
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Know that rational numbers and real numbers can be ordered and that the complex numbers cannot be ordered, but that any polynomial equation wit hreal coefficients can be solved in the complex field
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What does it mean ato be ordered?
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totally ordered: any two elements a and b in set, a <- b or b<- a.
Because of the Trichotomy Axiom of real numbers, we know that any subset of real numbers is ordered. |
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Trichotomy Axiom of real numbers
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if x and y are real numbers
i. x<y ii. x>y iii. x = y |
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Why can't complex numbers be ordered
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X
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Fundamental Theorem of Algebra
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if f(x0 is a polynomial with real coefficients, then f(x) can be factored into linear and quadratic factors, with real coefficients.
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Every complex polynomial has a root in C. This is a fancy way of saying what?
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C is an "algebraically closed" field
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Which of the following sets is a nordered field?
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Rational numbers
integers, natural numbers are all ordered <, because they are subsets of the ordered real numbers |
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List 3 reasons why the set of 2 by 2 matricies with real number entries do not form an ordered field.
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1. hard to order them in a meaningful way
2. set of 2 by 2 real matricies do not form a field 3. matricies like (2x2) ; (1 0 , 0 0 ) do not even have a multiplicative inverse. |
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What is the maximum number of complex solutions to x^17 - 573x^9 + 54 x^8 - 167x + 2 =0?
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17, the only time ther emay be fewer than 17 complex roots is if some of the roots have a multiplicity greater than 1 (double roots, triple roots, etc.)
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5d, 5e is really challenging. Do some research about this one
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z
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Rational Root Theorem Proof
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z
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What is the Factor Theorem?
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says that (x-b) is a factor of f(x) if and only if f(b) = 0.
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What is the Remainder Theorem?
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z
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How do you use the Remainder THeorem with synthetic substitution?
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z
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What is the Conjugate Roots Theorem for poplynomial equations with real coefficients?
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if f(x) is a polynomial with real coefficients, and if F(a + bi) = 0, then the Conjgate Roots Theorem says that f(a=bi) = 0
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What is the conjugate ROots Theorem for polynomial equations wit hrational coefficients?
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pg. 17
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What is the Quadratic Formula for real and complex quadratic polynomials?
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pg. 17
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What is the Binomial Theorem?
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pg. 17
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Show that in x^2 + bx + c =0, the sum of the two roots is -b and the product of the two roots is c.
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pg. 19
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Solve z^2 - iz + 2 =0
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Let 2x^4 - x^3 - 20x^2 + 13x + 30
i. list all possible rational roots ii. find all rational roots. iii. find all roots |
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Let 6x^4 + 7x^3 + 6x^2 - 1 = 0
i. list all possible rational roots ii. find all rational roots. iii. find all roots |
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Factor x^3 - x - 6 if you know that one root is -1 +i sqroot(2)
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pg. 19
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Derive the quadratic Formula
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pg 20
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Explain why the number of waqys to choose k objects from a group of n is the same as the number of ways to choose n-k objects from a group of n.
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If you choose k objects to include in your subgroup, then you could also think that simultaneously choosing n-k objects to exclude from your subgroup.
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What is the fundamental Theorem of Algebra
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It says that if f(x) is a polynomial with real coefficients, then f(x) can be factored into linear and quadratic factors, each with real coefficients
Could also have complex coefficients |
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How do yo use the Fundamental Theorem of Algebra to analyze polynomial equations?
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when determining the number of roots a polynomial has.
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What is a relation?
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a relation from a set A to a set B is a set of ordered pairs (x,y), where x e A and y e B
A relation on the real numbers is a subset R x R = R^2 |
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What is a function? What are domain and range?
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A function f from A to B is a relation from A to B that satisfies:
i. for every element x e A, there is an ordered pair (x,y )e f (establishes existence) ii. if (x,y) e f and (x,z) e f, then y = z (establishes uniqueness) The set A is called the domain of f The range of f is NOT the set B, but rather {f(x) : x e A} u- B. The set B is called the codomain of f |
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What is a one-to-one function?
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a function f : a > B is one to one if for all b e B, there is at most one x e A satisfying f(x) =b.
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What is an onto function?
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each element in the codomain has at least one element mapping to it
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To shift the graph of y - f*x) up by k units you ____ k to the ______ of the function f. The new graph is y = ______.
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add, output (or y value) , f(x) + k
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To shift the graph y = f(x) right by k units, you _____ k to the ____ of the function f. The new graph is y = ______.
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subtract, input (x-value), f (x-k)
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To stretch the graph of y = f(x) vertically by a factor of d units, you ____ the ____ of the function f by d. The new graph is y = ____.
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multiply, output, df(x)
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To Stretch the graph of y = f(x) horizontally by a factor of d units, you _____ the ____ of the function f by d. The new graph is y = _____.
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ANS: divide, input, f(x/d).
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TO reflect the graph of y - f(x) over the x-axis, you ____ the _____ of the function f by -1. The new graph is y = ______.
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multiply, output-f(x)
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identity function
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The composition of a function and its inverse should be the identity function (inverse "undoes" whatever the original function does.
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What does it mean if the identity function is "inert" under composition?
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the composition of any function g with the identity function is equal ot the function g.
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What is a vector?
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A vector is a math object that has a magnitude and direction.
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dot product?
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v . W = v1w1 + v2w2+....
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