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1st year Maths > VGLA sets, Matrices and Vectors > Flashcards

VGLA sets, Matrices and Vectors Flashcards

1
Q

set

A

a collection of things

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2
Q

|A| (Sets)

A

the order of A (how many elements A has)

Do not count repeated elements twice

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3
Q

[]

A

includes that last element

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4
Q

()

A

excludes that last element

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5
Q

empty dot

A

excludes the last value

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6
Q

full dot

A

includes the last value

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7
Q

A\B

A

(A-B)

those elements which are in A but not in B

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8
Q

Cartesian Product words

A

The collection of ordered pairs obtained by the product of two non empty sets.
IT RETURNS A SET FROM MULTIPLE SETS.

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9
Q

U

A

union (inclusive or)

or the universal set

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10
Q

∩

A

intersection (and)

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11
Q

a set of the empty set is not…

A

the empty set

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12
Q

exclusive or

A

it can either be one or the other but not both

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13
Q

inclusive or

A

it can be one or the other or both

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14
Q

set operations are commutative

A

the order of operations does not matter (except for A\B != B\A)

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15
Q

set operations are Associative

A

placement of brackets does not matter

AUB)UC = AU(BUC

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16
Q

set operations are distrubutive

A 
AU(B∩C) = (AUB)∩(AUC)
A∩(BUC) = (A∩B)U(A∩C)
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17
Q

A’

A

complement of A (everything but the elements in A)

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18
Q

De Morgans Law

A 
(AUB)' = A' ∩ B'
(A∩B)' = A' U B'
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19
Q

Cartesian Product

A

AxB = {(a,b): aЄA, bЄB}

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20
Q

induction steps

A
  1. show true when n=1
  2. assume true when n=k
  3. show true when n=k+1
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21
Q

ℕ

A

natural numbers, positive integers

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22
Q

Z

A

integers

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23
Q

Q

A

rational numbers

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24
Q

What makes equations linear?

A

no products + no powers

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25
Q

system of linear equations

A

a collection of equations that are related, not every equation has to have all the unknowns.

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26
Q

When does a system of linear equations have a solution (c1, c2, c3…)

A

whenever the values x1 = c1, x2 = c2 etc simultaneously satisfy all the equations

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27
Q

possible solutions for systems of equations:

A
  • no solutions
  • a unique solution
  • infinitely many solutions
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28
Q

homogeneous system

A

The constants (RHS) are all equal to 0

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29
Q

solution of a homogeneous system

A

(0,0,0) (trivial solution, not often of interest)

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30
Q

matrix

A

and array of numbers

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31
Q

mxn order matrix

A

m rows and n columns

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32
Q

aij

A

access the element of a matrix where i is the row and j is the column

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33
Q

two matrices are equal iff

A
  • same order

- every element is equal

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34
Q

sub matrix

A

part of a bigger matrix (not equal to the original matrix)

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35
Q

Rule for matrix addition

A

matrices must be the same size

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36
Q

zero matrix

A

a matrix where all the elements are 0

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37
Q

properties of matrix addition: closed

A

if you add two matrices of the same order the output has the same order

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38
Q

properties of matrix addition: Associativity

A

the placement of brackets doesn’t matter

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39
Q

properties of matrix addition: Identity

A

A + the zero matrix = A

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40
Q

properties of matrix addition: Inverse

A

A + its inverse = the zero matrix

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41
Q

properties of matrix addition: commutativity

A

order of addition doesn’t matter

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42
Q

multiplying a mxn matrix by a nxp matrix

A

a matrix of size mxp

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43
Q

properties of matrix multiplication:closed

A

if you take any square matrices the result will be in the original subset

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44
Q

properties of matrix multiplication: Associativity

A

placement of brackets doesn’t matter

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45
Q

properties of matrix multiplication: Identity

A

IA = AI = A

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46
Q

properties of matrix multiplication: Inverse

A

not possible

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47
Q

properties of matrix multiplication: Commutative

A

A.B != B.A

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48
Q

diagonal matrix

A

aij = 0 for i!=j

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49
Q

Upper triangular matrix

A

numbers in top triangle.

aij = 0 where i>j

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50
Q

Lower triangular matrix

A

numbers in bottom triangle

aij = o where i

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51
Q

rules for inverse matrices

A
  • a matrix can only have one inverse

- only for square matrices

52
Q

EROs (doesn’t matter in a system of equations)

A
  • swapping rows
  • taking a multiple of a row
  • Adding a multiple of another row
53
Q

row echelon form

A
  • all rows consisting of only 0 are at the bottom
  • successive non 0 rows have more 0s all left of the 1
  • the first non 0 number in any row is 1
54
Q

write solution sets

A

{(x,y,z)}

55
Q

EROs

A
  • swapping rows
  • taking a multiple of a row
  • Adding a multiple of another row
56
Q

two sets are equal iff

A

A⊆B and B⊆A

57
Q

solutions of equations in echelon form

A
  • if 0 = n (n is any non zero number) then no solutions
  • if 0=0 then infinitely many solutions
  • else a unique solution
58
Q

Gaussian Elimination

A

the process of converting the augmented matrix to echelon form then solving and finding all the possible solutions

59
Q

systems of equations with three unknowns described as planes in space

A
  1. all coicident (all the same plane, infinite solutions)
  2. all parallel (no solutions)
  3. two parallel planes + one intersecting plane (gives two lines of intersection, no solution)
  4. intersect at a single point (unique solution)
  5. intersect along a line ( infinite solutions)
  6. do not intersect at a single point or line (no solutions)
60
Q

(A.B)^-1 =

A

B^-1.A^-1

61
Q

Inverse of a 2x2 matrix

A

Let A = ( a b c d)

A^-1 = 1/ad-bc(d -b -c a)

62
Q

determinant of a 2x2 matrix

A

ad-bc

63
Q

If finding the inverse through the augmented matrix, an inverse does not exist if…

A

you can’t form the identity matrix on the LHS

64
Q

Elementary matrix

A

a matrix that differs from the identity matrix by one single row operation

65
Q

Eᵢⱼ

A

An elementary matrix where the rows i and j are swapped

66
Q

Eᵢ(λ)

A

An elementary matrix where one row is multiplied by the constant

67
Q

Eᵢⱼ(λ)

A

A matrix where one row is replaced by a sum of two rows

68
Q

An elementary matrix cannot…

A

represent 2 row operations at the same time

69
Q

inverse of Eᵢⱼ

A

itself

70
Q

inverse of Eᵢ(λ)

A

The inverse is Eᵢ(1/λ)

71
Q

inverse of Eᵢⱼ(λ)

A

The inverse is Eᵢⱼ(-λ)

72
Q

vector quantity

A

a value that has both magnitude and direction.

73
Q

vector examples

A

displacement, velocity, acceleration

74
Q

vector a

A

the subset of all the line segments with the same length and direction

75
Q

two vectors are equal if and only if

A

their magnitude and direction are equal

76
Q

properties of vector addition

A
  1. Internal (closed)
  2. Associative
  3. Identity (Zero matrix)
  4. inverse (-)
  5. commutative
77
Q

scalar multiple

A

where each element of a vector is multiplied by a constant α

78
Q 
vector distributivity (+)
α,β are scalars and u,v vectors
A

(α+β)v = αv + βv
AND
α(u+v)=αu+αv

79
Q

vector mixed associativty

A

α(βv)=(αβ)v

80
Q

vector examples

A

displacement, velocity, acceleration

81
Q

vector a

A

the subset of all the line segments with the same length and direction

82
Q

two vectors are equal if and only if

A

their magnitude and direction are equal

83
Q

properties of vector addition

A
  1. Internal (closed)
  2. Associative
  3. Identity (Zero matrix)
  4. inverse (-)
  5. commutative
84
Q

scalar multiple

A

where each element of a vector is multiplied by a constant α

85
Q

vector distributivity

α,β are scalars and u,v vectors

A

(α+β)v = αv + βv
AND
α(u+v)=αu+αv

86
Q

length of a position vector in terms of components

A

√(x²+y²+z²)

87
Q

length of a (non position) vector in terms of componets

A

√((x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²)

88
Q

unit vector

A

any vector with magnitude 1

89
Q

collinear vectors

A

the vectors lie on the same line or on parallel lines

90
Q

Scalar/Dot product definition

A

u,v!=0
u.v = |u||v|cosθ
u,v=0 (either)
u.v=0

91
Q

orthogonal

A

vectors are perpendicular

92
Q

orthonormal

A

vectors are perpendicular and have magnitude 1

93
Q

if u and v are perpendicular then u.v =

A

0

94
Q

length of a (non position) vector in terms of componets

A

√((x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²)

95
Q

|u| = √(u.u) reasonining

A

θ=0
u.u=|u||u|
u.u = |u|²
|u| = √(u.u)

96
Q

vector equation of a line

A

r = p + αq

where p is a position vector and q is a direction vector

97
Q

Scalar/Dot product definition

A

u,v!=0
u.v = |u||v|cosθ
u,v=0 (either)
u.v=0

98
Q

properties of the dot/scalar porduct

A
  1. not internal (no closed)
  2. no associative
  3. no identity
  4. no inverse
  5. commutative
99
Q

how to calculate the dot product from component form vectors

A

u.v = x₂x₁+y₂y₁+z₂z₁

100
Q

projection of v unto u

A

projᵤV = (v.u/|u|)i/|u|

101
Q

Use of right hand rule

A

to find the direction of vxu and curl your finger from v to u, which ever way your thumb points in the direction of vxu

102
Q

vector product

A

u,v!=0 and u,v are no parallel
|uxv|=|u||v|sinθ
The direction of uxv is perpendicular to both u and v

103
Q 
vector distributivity (.)
u,v,w are vectors
A

u.(v+w) = u.v + u.w

104
Q

u.(αv) =

A

α(v.u.)

105
Q

how to calculate the dot product from component form vectors

A

u.v = x₂x₁+y₂y₁+z₂z₁

106
Q

Associativity of vector product

A

vx(αu) = α(vxu)

107
Q

vector product of vectors in component form

A

uxv = (x₁z₂-z₁y₂, z₁x₂-x₁z₂, x₁y₂-y₁x₂)

108
Q

vector product

A

u,v!=0 and u,v are no parallel
|uxv|=|u||v|sinθ
The direction of uxv is perpendicular to both u and v

109
Q

properties of vector product

A
  1. Internal, closed
  2. not associative
  3. no identity
  4. no inverse
  5. not commutative
110
Q

normal vector

A

a non zero vector n is a normal vector to a plane if every directed line segment representing n is perpendicular tp the plane

111
Q

Distributivity of vector product

A

vx(v+w) = vxv +vxw

112
Q

Associativity of vector product

A

vx(αu) = α(vxu)

113
Q

vector product of vectors in component form

A

uxv = (x₁z₂-z₁y₂, z₁x₂-x₁z₂, x₁y₂-y₁x₂)

114
Q

distance between the point and a plane

A

d(p,π) = n.p-d/|n|

115
Q

vector equation of a plane

A

r.n = p.n

116
Q

normal vector

A

a non zero vector n is a normal vector to a plane if every directed line segment representing n is perpendicular

117
Q

cartesian equation of a plane

A

ax + by + cz = d

118
Q

intersection of planes

A
  1. all coincident (Infinite)
  2. all parallel (none)
  3. two parallel + one intersecting (none)
  4. intersect at a single point (unique)
  5. intersect along a line (infinite)
  6. form a triangle (none)
119
Q

interaction between lines and planes

A
  1. line intersects plane (unique solution)
  2. line paralell to plane (no solution)
  3. line in plane (inifinite solutions)
120
Q

distance between the point and a plane

A

d(p,π) = n.p-d/|n|

121
Q

distance between two parallel planes

A

d(π₁,π₂) = (d₁-d₂)/|n|

122
Q

interaction between two lines in space

A
  1. parallel lines
  2. intersection
  3. be skew
123
Q

angle between two lines in space

A

the angle between their direction vectors

cosθ = v.u/|v||u|

124
Q

Scalar triple product

A

u.(vxw)

125
Q

what does the scalar triple product give geonmetrically

A

the volume of the parallelipipied

126
Q

uxvxw

A

undefined (vector product is not associative)

1st year Maths (24 decks)

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