VGLA sets, Matrices and Vectors Flashcards
1
Q
set
A
a collection of things
2
Q
|A| (Sets)
A
the order of A (how many elements A has)
Do not count repeated elements twice
3
Q
[]
A
includes that last element
4
Q
()
A
excludes that last element
5
Q
empty dot
A
excludes the last value
6
Q
full dot
A
includes the last value
7
Q
A\B
A
(A-B)
those elements which are in A but not in B
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.
9
Q
U
A
union (inclusive or)
or the universal set
10
Q
∩
A
intersection (and)
11
Q
a set of the empty set is not…
A
the empty set
12
Q
exclusive or
A
it can either be one or the other but not both
13
Q
inclusive or
A
it can be one or the other or both
14
Q
set operations are commutative
A
the order of operations does not matter (except for A\B != B\A)
15
Q
set operations are Associative
A
placement of brackets does not matter
AUB)UC = AU(BUC
16
Q
set operations are distrubutive
A
AU(B∩C) = (AUB)∩(AUC) A∩(BUC) = (A∩B)U(A∩C)
17
Q
A’
A
complement of A (everything but the elements in A)
18
Q
De Morgans Law
A
(AUB)' = A' ∩ B' (A∩B)' = A' U B'
19
Q
Cartesian Product
A
AxB = {(a,b): aЄA, bЄB}
20
Q
induction steps
A
- show true when n=1
- assume true when n=k
- show true when n=k+1
21
Q
ℕ
A
natural numbers, positive integers
22
Q
Z
A
integers
23
Q
Q
A
rational numbers
24
Q
What makes equations linear?
A
no products + no powers
25
system of linear equations
a collection of equations that are related, not every equation has to have all the unknowns.
26
When does a system of linear equations have a solution (c1, c2, c3...)
whenever the values x1 = c1, x2 = c2 etc simultaneously satisfy all the equations
27
possible solutions for systems of equations:
- no solutions
- a unique solution
- infinitely many solutions
28
homogeneous system
The constants (RHS) are all equal to 0
29
solution of a homogeneous system
(0,0,0) (trivial solution, not often of interest)
30
matrix
and array of numbers
31
mxn order matrix
m rows and n columns
32
aij
access the element of a matrix where i is the row and j is the column
33
two matrices are equal iff
- same order
| - every element is equal
34
sub matrix
part of a bigger matrix (not equal to the original matrix)
35
Rule for matrix addition
matrices must be the same size
36
zero matrix
a matrix where all the elements are 0
37
properties of matrix addition: closed
if you add two matrices of the same order the output has the same order
38
properties of matrix addition: Associativity
the placement of brackets doesn't matter
39
properties of matrix addition: Identity
A + the zero matrix = A
40
properties of matrix addition: Inverse
A + its inverse = the zero matrix
41
properties of matrix addition: commutativity
order of addition doesn't matter
42
multiplying a mxn matrix by a nxp matrix
a matrix of size mxp
43
properties of matrix multiplication:closed
if you take any square matrices the result will be in the original subset
44
properties of matrix multiplication: Associativity
placement of brackets doesn't matter
45
properties of matrix multiplication: Identity
I*A = A*I = A
46
properties of matrix multiplication: Inverse
not possible
47
properties of matrix multiplication: Commutative
A.B != B.A
48
diagonal matrix
aij = 0 for i!=j
49
Upper triangular matrix
numbers in top triangle.
| aij = 0 where i>j
50
Lower triangular matrix
numbers in bottom triangle
| aij = o where i
51
rules for inverse matrices
- a matrix can only have one inverse
| - only for square matrices
52
EROs (doesn't matter in a system of equations)
- swapping rows
- taking a multiple of a row
- Adding a multiple of another row
53
row echelon form
- 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
write solution sets
{(x,y,z)}
55
EROs
- swapping rows
- taking a multiple of a row
- Adding a multiple of another row
56
two sets are equal iff
A⊆B and B⊆A
57
solutions of equations in echelon form
- if 0 = n (n is any non zero number) then no solutions
- if 0=0 then infinitely many solutions
- else a unique solution
58
Gaussian Elimination
the process of converting the augmented matrix to echelon form then solving and finding all the possible solutions
59
systems of equations with three unknowns described as planes in space
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
(A.B)^-1 =
B^-1.A^-1
61
Inverse of a 2x2 matrix
Let A = ( a b c d)
| A^-1 = 1/ad-bc(d -b -c a)
62
determinant of a 2x2 matrix
ad-bc
63
If finding the inverse through the augmented matrix, an inverse does not exist if...
you can't form the identity matrix on the LHS
64
Elementary matrix
a matrix that differs from the identity matrix by one single row operation
65
Eᵢⱼ
An elementary matrix where the rows i and j are swapped
66
Eᵢ(λ)
An elementary matrix where one row is multiplied by the constant
67
Eᵢⱼ(λ)
A matrix where one row is replaced by a sum of two rows
68
An elementary matrix cannot...
represent 2 row operations at the same time
69
inverse of Eᵢⱼ
itself
70
inverse of Eᵢ(λ)
The inverse is Eᵢ(1/λ)
71
inverse of Eᵢⱼ(λ)
The inverse is Eᵢⱼ(-λ)
72
vector quantity
a value that has both magnitude and direction.
73
vector examples
displacement, velocity, acceleration
74
vector a
the subset of all the line segments with the same length and direction
75
two vectors are equal if and only if
their magnitude and direction are equal
76
properties of vector addition
1. Internal (closed)
2. Associative
3. Identity (Zero matrix)
4. inverse (-)
5. commutative
77
scalar multiple
where each element of a vector is multiplied by a constant α
78
```
vector distributivity (+)
α,β are scalars and u,v vectors
```
(α+β)v = αv + βv
AND
α(u+v)=αu+αv
79
vector mixed associativty
α(βv)=(αβ)v
80
vector examples
displacement, velocity, acceleration
81
vector a
the subset of all the line segments with the same length and direction
82
two vectors are equal if and only if
their magnitude and direction are equal
83
properties of vector addition
1. Internal (closed)
2. Associative
3. Identity (Zero matrix)
4. inverse (-)
5. commutative
84
scalar multiple
where each element of a vector is multiplied by a constant α
85
vector distributivity
| α,β are scalars and u,v vectors
(α+β)v = αv + βv
AND
α(u+v)=αu+αv
86
length of a position vector in terms of components
√(x²+y²+z²)
87
length of a (non position) vector in terms of componets
√((x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²)
88
unit vector
any vector with magnitude 1
89
collinear vectors
the vectors lie on the same line or on parallel lines
90
Scalar/Dot product definition
u,v!=0
u.v = |u||v|cosθ
u,v=0 (either)
u.v=0
91
orthogonal
vectors are perpendicular
92
orthonormal
vectors are perpendicular and have magnitude 1
93
if u and v are perpendicular then u.v =
0
94
length of a (non position) vector in terms of componets
√((x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²)
95
|u| = √(u.u) reasonining
θ=0
u.u=|u||u|
u.u = |u|²
|u| = √(u.u)
96
vector equation of a line
r = p + αq
| where p is a position vector and q is a direction vector
97
Scalar/Dot product definition
u,v!=0
u.v = |u||v|cosθ
u,v=0 (either)
u.v=0
98
properties of the dot/scalar porduct
1. not internal (no closed)
2. no associative
3. no identity
4. no inverse
5. commutative
99
how to calculate the dot product from component form vectors
u.v = x₂x₁+y₂y₁+z₂z₁
100
projection of v unto u
projᵤV = (v.u/|u|)i/|u|
101
Use of right hand rule
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
vector product
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
```
vector distributivity (.)
u,v,w are vectors
```
u.(v+w) = u.v + u.w
104
u.(αv) =
α(v.u.)
105
how to calculate the dot product from component form vectors
u.v = x₂x₁+y₂y₁+z₂z₁
106
Associativity of vector product
vx(αu) = α(vxu)
107
vector product of vectors in component form
uxv = (x₁z₂-z₁y₂, z₁x₂-x₁z₂, x₁y₂-y₁x₂)
108
vector product
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
properties of vector product
1. Internal, closed
2. not associative
3. no identity
4. no inverse
5. not commutative
110
normal vector
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
Distributivity of vector product
vx(v+w) = vxv +vxw
112
Associativity of vector product
vx(αu) = α(vxu)
113
vector product of vectors in component form
uxv = (x₁z₂-z₁y₂, z₁x₂-x₁z₂, x₁y₂-y₁x₂)
114
distance between the point and a plane
d(p,π) = n.p-d/|n|
115
vector equation of a plane
r.n = p.n
116
normal vector
a non zero vector n is a normal vector to a plane if every directed line segment representing n is perpendicular
117
cartesian equation of a plane
ax + by + cz = d
118
intersection of planes
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
interaction between lines and planes
1. line intersects plane (unique solution)
2. line paralell to plane (no solution)
3. line in plane (inifinite solutions)
120
distance between the point and a plane
d(p,π) = n.p-d/|n|
121
distance between two parallel planes
d(π₁,π₂) = (d₁-d₂)/|n|
122
interaction between two lines in space
1. parallel lines
2. intersection
3. be skew
123
angle between two lines in space
the angle between their direction vectors
| cosθ = v.u/|v||u|
124
Scalar triple product
u.(vxw)
125
what does the scalar triple product give geonmetrically
the volume of the parallelipipied
126
uxvxw
undefined (vector product is not associative)
