math Flashcards
degree of polynomial for x^5 + 4x^2
5
in gauss elimination how to determine if inf many solutions exist and no soulutions exist
a row is 0+0+0 = 0 for inf many
a row is 0+0+0 = 12 for no soultions
distance between two points
square root of (x2-x1)^2 + (y2-y1)^+(z2-z1)^2
radius equation for a sphere with centre (a,b,c)
(x-a)^ + (y-b)^2 + (z-c)^2 =r^2
what does number of elements in a vector show
dimension/order
magnitude of a vector or norm
||a|| = square root a1^2 + a2^2 + an^2
how to find dot product between two vectors
Transpose of vector a * vector b
orthogonal vector
dot proudct is 0
showing independent vectors
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
Angle between vectors
cos0 = a^transpoes * b / ||a|| * ||b||
scalar projection a onto b
a^T *b / ||b||
vector projection a onto b
(a^T * b / ||b||^2) b
unit vector of b
b/||b||
proof for orthagonal
if vectors a and b are ortogonal, the angle is 90 and cos angle = 0
hence
a * b = ||a|| * ||b|| * cos0 = 0
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
trace of a matrix
sum of main diagonal a11, a22, a nn give us trace
Condition to multiply two matricies
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
what does it mean for matrxi to be symmetric
A = A^T only for square matrix
how to find determinant of 2x2 matrix
[a11, a12]
[a21, a22]
a11a22 - a12a21
how to find determinant of 3x3 matrix
extend the matrix with the first 2 column
how to find determinant using cofactor method
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
How to find inverse
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
inverse of 2x2 trick
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]
A matrix A of dimension m × n has rank(A) ≤ min(m, n).
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
Craners Rule
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
how to find rank of a matrix
number of non zero rows at the end of gauss elimination
what is the rank of matrix n x n with det != 0
n
rank of matrix mx n
rank min(m,n)
how to find inverse with gauss jordan
combine matrix A with the idenitiy matrix100
010
001
then you need the idenity matrix on the left side
what does singular mean
det = 0
format for eigenvector problems with matrix
[5,-2]
[9,-6]
(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
rules for a group (G,*)
group (G,)
operation * closed on G - if a,b in G then (ab) in G
* is associate- if a,b,c in G then (ab)c = a(bc)
G contains identity of operation * - where e is identity, ae = ea = a
Inverses exist for every element in set- yx = e, xy = e where e is identity
Monoid for (G,*)
- closed on G a*b in G
- associate - (ab)c = a(bc)
G contains identity * ae = ea = a
semiGroup rules(G,*)
- closed on G a*b in G
- associate - (ab)c = a(bc)
Abelian group
if its operation is commutative and also group
(ab) = ba
rings rules( why are we doing this?)
consist of 2 binary operations (R,,o)
* and o are closed (ab) and (a o b) in R
* and o is associative- (xy)z = x(yz)
* is commutative xy = yx
R has identity of * - xe = ex = x
Inverses exists for * (xy) = yx = 0
o is distributive with respect to - if x,y,z in R then it has x o (yz) =
(x o y) * (x o z)
(x*y) o z = (x o z) * (y o z)
Field
same as ring but also both operations are commutative and inverses and identity exist
what is order of a group
2^n = identity
What is the probability of E if the sample space S is) of equally likely outcomes and e is a subset of S
|E|/|S|
- Probability consequences P(∅) = 0
- if E subset F then P(E) <= P(F)
- P(EUF) = P(E) + P(F) - P(E^F)
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)
what does independent events mean
P(E^F) = P(E) * P(F)
Conditional probability formula P(A\B)
P(A^B)/P(B)
Total probability theorm??
Bayes Theorm
A and B events with non zero probability
Say P(A/B) =
(P(B/A) x P(A)) / P(B)
probability mass function
probability mean
x1P(x1) + x2P(x2)+ xnP(xn)
cumulatitve distribution function
F(b) = P(X<=b)
so F(1) = P(x=0) + P(x=1)
Probability variance
Sum of i = 1 to n (xi^2 *P(xi)) - μ^2
Bernoulli distribution
how it works and mean and variance
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)
Binomial Distribution
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
binomial distribution mean and variance and s.d
np
np*q
Geometric distribution
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?
geometric mean and variance
μ = 1/p
var = q/p^2
Poisson Distribution
mean and variance
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
Standard normal equation
Z = x- μ / σ
Z~N[0,1]
0 is mean
1 is variance
how would u solve these
P(0<Z<0.13)
P(Z<0.23)
P(Z>0.33)
P(0<Z<0.135)
P(0.23<Z<0.33)
- look in table
- look for P(0<Z<0.23) in table = 0.091 then add on P(Z<0) so 0.5 + 0.091
- 0.5 - P(0<Z<0.33)
4.P of 0.13 + 0.5(P(0.14) -P(0.13)) - P(0.33) - P(0.23)
remember converting snd from nd
what is sample mean distribution where there are multiple random sample variables
_
X ~ N[μ,σ^2/N)
Arithmetic mean
Geometric mean(values with different properties)
Harmonic mean(avg of rates) (same distance different speed)
Weighted ,eam
x1 + x2 + xn / n
root n sqrt(X1X2Xn)
n/(1/x) + (1/x2)
W1X1 + W2X2/ W1 + W2
