VGLA sets, Matrices and Vectors Flashcards
set
a collection of things
|A| (Sets)
the order of A (how many elements A has)
Do not count repeated elements twice
[]
includes that last element
()
excludes that last element
empty dot
excludes the last value
full dot
includes the last value
A\B
(A-B)
those elements which are in A but not in B
Cartesian Product words
The collection of ordered pairs obtained by the product of two non empty sets.
IT RETURNS A SET FROM MULTIPLE SETS.
U
union (inclusive or)
or the universal set
∩
intersection (and)
a set of the empty set is not…
the empty set
exclusive or
it can either be one or the other but not both
inclusive or
it can be one or the other or both
set operations are commutative
the order of operations does not matter (except for A\B != B\A)
set operations are Associative
placement of brackets does not matter
AUB)UC = AU(BUC
set operations are distrubutive
AU(B∩C) = (AUB)∩(AUC) A∩(BUC) = (A∩B)U(A∩C)
A’
complement of A (everything but the elements in A)
De Morgans Law
(AUB)' = A' ∩ B' (A∩B)' = A' U B'
Cartesian Product
AxB = {(a,b): aЄA, bЄB}
induction steps
- show true when n=1
- assume true when n=k
- show true when n=k+1
ℕ
natural numbers, positive integers
Z
integers
Q
rational numbers
What makes equations linear?
no products + no powers
system of linear equations
a collection of equations that are related, not every equation has to have all the unknowns.
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
possible solutions for systems of equations:
- no solutions
- a unique solution
- infinitely many solutions
homogeneous system
The constants (RHS) are all equal to 0
solution of a homogeneous system
(0,0,0) (trivial solution, not often of interest)
matrix
and array of numbers
mxn order matrix
m rows and n columns
aij
access the element of a matrix where i is the row and j is the column
two matrices are equal iff
- same order
- every element is equal
sub matrix
part of a bigger matrix (not equal to the original matrix)
Rule for matrix addition
matrices must be the same size
zero matrix
a matrix where all the elements are 0
properties of matrix addition: closed
if you add two matrices of the same order the output has the same order
properties of matrix addition: Associativity
the placement of brackets doesn’t matter
properties of matrix addition: Identity
A + the zero matrix = A
properties of matrix addition: Inverse
A + its inverse = the zero matrix
properties of matrix addition: commutativity
order of addition doesn’t matter
multiplying a mxn matrix by a nxp matrix
a matrix of size mxp
properties of matrix multiplication:closed
if you take any square matrices the result will be in the original subset
properties of matrix multiplication: Associativity
placement of brackets doesn’t matter
properties of matrix multiplication: Identity
IA = AI = A
properties of matrix multiplication: Inverse
not possible
properties of matrix multiplication: Commutative
A.B != B.A
diagonal matrix
aij = 0 for i!=j
Upper triangular matrix
numbers in top triangle.
aij = 0 where i>j
Lower triangular matrix
numbers in bottom triangle
aij = o where i
rules for inverse matrices
- a matrix can only have one inverse
- only for square matrices
EROs (doesn’t matter in a system of equations)
- swapping rows
- taking a multiple of a row
- Adding a multiple of another row
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
write solution sets
{(x,y,z)}
EROs
- swapping rows
- taking a multiple of a row
- Adding a multiple of another row
two sets are equal iff
A⊆B and B⊆A
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
Gaussian Elimination
the process of converting the augmented matrix to echelon form then solving and finding all the possible solutions
systems of equations with three unknowns described as planes in space
- all coicident (all the same plane, infinite solutions)
- all parallel (no solutions)
- two parallel planes + one intersecting plane (gives two lines of intersection, no solution)
- intersect at a single point (unique solution)
- intersect along a line ( infinite solutions)
- do not intersect at a single point or line (no solutions)
(A.B)^-1 =
B^-1.A^-1
Inverse of a 2x2 matrix
Let A = ( a b c d)
A^-1 = 1/ad-bc(d -b -c a)
determinant of a 2x2 matrix
ad-bc
If finding the inverse through the augmented matrix, an inverse does not exist if…
you can’t form the identity matrix on the LHS
Elementary matrix
a matrix that differs from the identity matrix by one single row operation
Eᵢⱼ
An elementary matrix where the rows i and j are swapped
Eᵢ(λ)
An elementary matrix where one row is multiplied by the constant
Eᵢⱼ(λ)
A matrix where one row is replaced by a sum of two rows
An elementary matrix cannot…
represent 2 row operations at the same time
inverse of Eᵢⱼ
itself
inverse of Eᵢ(λ)
The inverse is Eᵢ(1/λ)
inverse of Eᵢⱼ(λ)
The inverse is Eᵢⱼ(-λ)
vector quantity
a value that has both magnitude and direction.
vector examples
displacement, velocity, acceleration
vector a
the subset of all the line segments with the same length and direction
two vectors are equal if and only if
their magnitude and direction are equal
properties of vector addition
- Internal (closed)
- Associative
- Identity (Zero matrix)
- inverse (-)
- commutative
scalar multiple
where each element of a vector is multiplied by a constant α
vector distributivity (+) α,β are scalars and u,v vectors
(α+β)v = αv + βv
AND
α(u+v)=αu+αv
vector mixed associativty
α(βv)=(αβ)v
vector examples
displacement, velocity, acceleration
vector a
the subset of all the line segments with the same length and direction
two vectors are equal if and only if
their magnitude and direction are equal
properties of vector addition
- Internal (closed)
- Associative
- Identity (Zero matrix)
- inverse (-)
- commutative
scalar multiple
where each element of a vector is multiplied by a constant α
vector distributivity
α,β are scalars and u,v vectors
(α+β)v = αv + βv
AND
α(u+v)=αu+αv
length of a position vector in terms of components
√(x²+y²+z²)
length of a (non position) vector in terms of componets
√((x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²)
unit vector
any vector with magnitude 1
collinear vectors
the vectors lie on the same line or on parallel lines
Scalar/Dot product definition
u,v!=0
u.v = |u||v|cosθ
u,v=0 (either)
u.v=0
orthogonal
vectors are perpendicular
orthonormal
vectors are perpendicular and have magnitude 1
if u and v are perpendicular then u.v =
0
length of a (non position) vector in terms of componets
√((x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²)
|u| = √(u.u) reasonining
θ=0
u.u=|u||u|
u.u = |u|²
|u| = √(u.u)
vector equation of a line
r = p + αq
where p is a position vector and q is a direction vector
Scalar/Dot product definition
u,v!=0
u.v = |u||v|cosθ
u,v=0 (either)
u.v=0
properties of the dot/scalar porduct
- not internal (no closed)
- no associative
- no identity
- no inverse
- commutative
how to calculate the dot product from component form vectors
u.v = x₂x₁+y₂y₁+z₂z₁
projection of v unto u
projᵤV = (v.u/|u|)i/|u|
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
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
vector distributivity (.) u,v,w are vectors
u.(v+w) = u.v + u.w
u.(αv) =
α(v.u.)
how to calculate the dot product from component form vectors
u.v = x₂x₁+y₂y₁+z₂z₁
Associativity of vector product
vx(αu) = α(vxu)
vector product of vectors in component form
uxv = (x₁z₂-z₁y₂, z₁x₂-x₁z₂, x₁y₂-y₁x₂)
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
properties of vector product
- Internal, closed
- not associative
- no identity
- no inverse
- not commutative
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
Distributivity of vector product
vx(v+w) = vxv +vxw
Associativity of vector product
vx(αu) = α(vxu)
vector product of vectors in component form
uxv = (x₁z₂-z₁y₂, z₁x₂-x₁z₂, x₁y₂-y₁x₂)
distance between the point and a plane
d(p,π) = n.p-d/|n|
vector equation of a plane
r.n = p.n
normal vector
a non zero vector n is a normal vector to a plane if every directed line segment representing n is perpendicular
cartesian equation of a plane
ax + by + cz = d
intersection of planes
- all coincident (Infinite)
- all parallel (none)
- two parallel + one intersecting (none)
- intersect at a single point (unique)
- intersect along a line (infinite)
- form a triangle (none)
interaction between lines and planes
- line intersects plane (unique solution)
- line paralell to plane (no solution)
- line in plane (inifinite solutions)
distance between the point and a plane
d(p,π) = n.p-d/|n|
distance between two parallel planes
d(π₁,π₂) = (d₁-d₂)/|n|
interaction between two lines in space
- parallel lines
- intersection
- be skew
angle between two lines in space
the angle between their direction vectors
cosθ = v.u/|v||u|
Scalar triple product
u.(vxw)
what does the scalar triple product give geonmetrically
the volume of the parallelipipied
uxvxw
undefined (vector product is not associative)
