math

Question 1

degree of polynomial for x^5 + 4x^2

Answer 1

5

Question 2

in gauss elimination how to determine if inf many solutions exist and no soulutions exist

Answer 2

a row is 0+0+0 = 0 for inf many
a row is 0+0+0 = 12 for no soultions

Question 3

distance between two points

Answer 3

square root of (x2-x1)^2 + (y2-y1)^+(z2-z1)^2

Question 4

radius equation for a sphere with centre (a,b,c)

Answer 4

(x-a)^ + (y-b)^2 + (z-c)^2 =r^2

Question 5

what does number of elements in a vector show

Answer 5

dimension/order

Question 6

magnitude of a vector or norm

Answer 6

||a|| = square root a1^2 + a2^2 + an^2

Question 7

how to find dot product between two vectors

Answer 7

Transpose of vector a * vector b

Question 8

orthogonal vector

Answer 8

dot proudct is 0

Question 9

showing independent vectors

Answer 9

the subspace cannot be larger then dimension of vector
for example 3 {c,d,e}
show that we cannot express one of them in terms of the other two.
use contradiction that in fact there exist integers m, n such that
c = m × d + n × e

once proved that cannot be express, have to show that d cannot be expressed by e

Question 10

Angle between vectors

Answer 10

cos0 = a^transpoes * b / ||a|| * ||b||

Question 11

scalar projection a onto b

Answer 11

a^T *b / ||b||

Question 12

vector projection a onto b

Answer 12

(a^T * b / ||b||^2) b

Question 13

unit vector of b

Answer 13

b/||b||

Question 14

proof for orthagonal

Answer 14

if vectors a and b are ortogonal, the angle is 90 and cos angle = 0
hence
a * b = ||a|| * ||b|| * cos0 = 0

Answer 15

A set of vectors in a vector space V is called a basis if the vectors are linearly
independent and every vector in the vector space is a linear combination of this set

Question 16

trace of a matrix

Answer 16

sum of main diagonal a11, a22, a nn give us trace

Question 17

Condition to multiply two matricies

Answer 17

Number of columns in first matrix must equal number of rows in second matrix.
then do row1 of first one x columns of second one

Question 18

what does it mean for matrxi to be symmetric

Answer 18

A = A^T only for square matrix

Question 19

how to find determinant of 2x2 matrix

Answer 19

[a11, a12]
[a21, a22]

a11a22 - a12a21

Question 20

how to find determinant of 3x3 matrix

Answer 20

extend the matrix with the first 2 column

Question 21

how to find determinant using cofactor method

Answer 21

for nxn matrix
[a11, a12, a13]
[a21, a,22, a23]
[a31, a,32, a33]

a11 * -1^2 * M11 + a21 * -1^3 * M21 + a31 * -1^4 * M31

Question 22

How to find inverse

Answer 22

determinant must be non zero

First Transpose the matrix.
Then find the Cofactor matrix of that including multipling each by -1^n+1
Multiply it by 1/determinant

Question 23

inverse of 2x2 trick

Answer 23

say D = [-4, -5]
[7 , -2]
first swap the numbers -4 amd -2
then change the sign of -5 and 7
so the determinant is 43 and the inverse is
1/43 * [-2, 5]
[-7, -4]

Question 24

A matrix A of dimension m × n has rank(A) ≤ min(m, n).

Answer 24

Proof. For a matrix A we have rank(A) ≤ m (the number of rows). But at the same time,
rank(A) = rank(AT
) ≤ n (the number of rows of AT

Question 25

Craners Rule

Answer 25

valid only for square matrix to solve for x or y or z xi = detAi/detA where Ai is obtained by replacing column vector with solution vector

Question 26

how to find rank of a matrix

Answer 26

number of non zero rows at the end of gauss elimination

Question 27

what is the rank of matrix n x n with det != 0

Answer 27

n

Question 28

rank of matrix mx n

Answer 28

rank min(m,n)

Question 29

how to find inverse with gauss jordan

Answer 29

combine matrix A with the idenitiy matrix100 010 001 then you need the idenity matrix on the left side

Question 30

what does singular mean

Answer 30

det = 0

Question 31

format for eigenvector problems with matrix [5,-2] [9,-6]

Answer 31

(5-λ)x1 -2x2 = 0 9x1 + (-6-λ)x2 = 0 then put in matix and solve for determinant then find values for λ so you get λ = -4 and λ = 3 put it in equation and solve for x1,x2 so 9x1 - 2x2 = 0 9/2x1 = x2 x1 is free variable so set x1 = x1 [x1] [9/2x1] for lamda = -4

Question 32

rules for a group (G,*)

Answer 32

group (G,*) operation * closed on G - if a,b in G then (a*b) in G * is associate- if a,b,c in G then (a*b)*c = a*(b*c) G contains identity of operation * - where e is identity, a*e = e*a = a Inverses exist for every element in set- y*x = e, x*y = e where e is identity

Question 33

Monoid for (G,*)

Answer 33

* closed on G a*b in G * associate - (a*b)*c = a*(b*c) G contains identity * a*e = e*a = a

Question 34

semiGroup rules(G,*)

Answer 34

* closed on G a*b in G * associate - (a*b)*c = a*(b*c)

Question 35

Abelian group

Answer 35

if its operation is commutative and also group (a*b) = b*a

Question 36

rings rules( why are we doing this?)

Answer 36

consist of 2 binary operations (R,*,o) * and o are closed (a*b) and (a o b) in R * and o is associative- (x*y)*z = x*(y*z) * is commutative x*y = y*x R has identity of * - x*e = e*x = x Inverses exists for * (x*y) = y*x = 0 o is distributive with respect to *- if x,y,z in R then it has x o (y*z) = (x o y) * (x o z) (x*y) o z = (x o z) * (y o z)

Question 37

Field

Answer 37

same as ring but also both operations are commutative and inverses and identity exist

Question 38

what is order of a group

Answer 38

2^n = identity

Question 39

What is the probability of E if the sample space S is) of equally likely outcomes and e is a subset of S

Answer 39

|E|/|S|

Question 40

1. Probability consequences P(∅) = 0 2. if E subset F then P(E) <= P(F) 3. P(EUF) = P(E) + P(F) - P(E^F)

Answer 40

P(∅) = 0 S U ∅ = S --> P(S U ∅) = P(S) P(S U ∅) = P(S) + P(∅) P(S) = P(S) + P(0) E U ( F-E) = F P(EU(F-E) = P(E) + P(F-E) = P(F) 0 <= P(F-E) (E^F) U (E-F) = E P(E) = P(E^F) + P(E-F) P(EUF) = P(E^F) + P(E-F) + P(F-E) P(EUF) = P(E) + P(F) - P(E^F)

Question 41

what does independent events mean

Answer 41

P(E^F) = P(E) * P(F)

Question 42

Conditional probability formula P(A\B)

Answer 42

P(A^B)/P(B)

Question 43

Total probability theorm??

Question 44

Bayes Theorm

Answer 44

A and B events with non zero probability Say P(A/B) = (P(B/A) x P(A)) / P(B)

Question 45

probability mass function

Question 46

probability mean

Answer 46

x1P(x1) + x2P(x2)+ xnP(xn)

Question 47

cumulatitve distribution function

Answer 47

F(b) = P(X<=b) so F(1) = P(x=0) + P(x=1)

Question 48

Probability variance

Answer 48

Sum of i = 1 to n (xi^2 *P(xi)) - μ^2

Question 49

Bernoulli distribution how it works and mean and variance

Answer 49

outcome is classified as only success or failure P(x=0) = 1-p =q P(x=1) = p μ = p variance = pq standard deviation sqrt(pq)

Question 50

Binomial Distribution

Answer 50

fixed # of observations. fixed probability n observations are independent each observation is success or failure x ~ B(n,p) P of x successes followed by n-x failures = p^x * q^n-x if we want any successes out of n tries (nCx) P^x * q^n-x

Question 51

binomial distribution mean and variance and s.d

Answer 51

n*p n*p*q

Question 52

Geometric distribution

Answer 52

Estimated number of events that need happeneing until we reach a success Probability of success = p and failure is 1-p = q probability of k-1 failures followed by success P(x=k) = q^k-1 * p e.g How many times expected to draw from a deck of cards until we get a 4. use mean. or A fair coin is flipped repeatedly. What is the probability that the first heads (success) occurs on the 4th flip?

Question 53

geometric mean and variance

Answer 53

μ = 1/p var = q/p^2

Question 54

Poisson Distribution mean and variance

Answer 54

Estimates number of successes happenning when we know the avg # of events that occur λ = avg number of events μ= λ variance= λ P(success) : P(x=k) = λ^k/k! * e^-λ

Question 55

Standard normal equation

Answer 55

Z = x- μ / σ Z~N[0,1] 0 is mean 1 is variance

Question 56

how would u solve these P(00.33) P(0

Answer 56

1. look in table 2. look for P(0

Question 57

remember converting snd from nd

Question 58

what is sample mean distribution where there are multiple random sample variables

Answer 58

_ X ~ N[μ,σ^2/N)

Question 59

Arithmetic mean Geometric mean(values with different properties) Harmonic mean(avg of rates) (same distance different speed) Weighted ,eam

Answer 59

x1 + x2 + xn / n root n sqrt(X1*X2*Xn) n/(1/x) + (1/x2) W1X1 + W2X2/ W1 + W2