VLGA definitions term 1 Flashcards
Set
A collection of things
order
how many elements a set has (do not count repeated elements twice)
cartesian product
the collection of ordered pairs obtained by the product of two non empty sets
difference (of A and B)
those elements which are in A but not in B
commutative
order of operations doesn’t matter
Associative
placement of brackets doesn’t matter
distributive
expansion of brackets
complement of A
All the elements in the universal set which are not in A
mathematical induction
if P(n) is a statement involving n∈ℕ (i) P(1) is true (ii) for each k∈ℕ, we have P(k) is true => P(k+1) is true Then P(n) is true for all n∈ℕ
linear equation
an equation that has no products of unknowns and no powers.
e.g. ax+by=c
solution to linear equations
the system has a solution whenever the values simultaneously satisfy all the equations
homogeneous
The constants in the system of equations are all equal to zero. There is a trivial solution that x₁, x₂, x₃, xₙ are equal to 0.
non homogeneous
The constants in the simultaneous equations are equal to anything
matrices are equal
if
(i) both have the same dimension mxn
(ii) aᵢⱼ = bᵢⱼ for all i,j
matrix
array of numbers
matrix order
mxn where m is the number of rows and n is the number of columns`
sub-matrix
a part of a bigger matrix
zero matrix
a matrix where all the elements are equal to zero
diagonal matrix
An nxn matrix is diagonal when:
aᵢⱼ = 0 for i!=j
identity matrix
A diagonal matrix where the the diagonal is all 1s
Upper Triangular matrix
any nxn matrix where aᵢⱼ = 0 where i>j
Lower Triangular matrix
any nxn matrix where aᵢⱼ = 0 where j>i
inverse matrix
Let A be an nxn matrix. If there exists an nxn matrix B such that A.B = I = B.A
Then is is said to be invertible and we write B = A⁻¹
B is called the inverse of A
singular
the matrix doesn’t have an inverse
Elementary row operation
an operation that does not change the solution set of linear equations:
- > swapping rows
- > taking multiples of a row
- > adding a multiple of a row
row echelon form
A matrix is in row echelon form when:
(i) all rows consisting of only zeros are at the bottom
(ii) First non-zero number in any row is 1
(iii) successive non-zero rows begin with more zeros left of the 1 then then rows above
Guassian Elimination
applying successive EROs to get the matrix into echelon form.
Elementary matrix
An nxn elementary matrix is obtained by performing one elementary row operation to the identity matrix I
vector quantity
a quantity has both magnitude and direction
scalar quantity
a quantity that has only magnitude
set of vectors in three dimensions
all three vectors together with the zero vector which has no magnitude or direction.
vector alternative
the subset of all line segments with the same length and direction
vector equality
Two vectors are equal if and only if they have the same magnitude and direction
Vector Addition
This is done in three cases:
(1) u!=0, v!= and u!=-v. Find R such that v = QR. Then u+v = PR
(2) suppose u!=0, v!=0 and u=-v. Then u+ v = 0
(3) if u or v is the zero vector, we set u+0=u
Vector Subtraction
Vector subtraction is defined as follows:
u-v = u+(-v)
Scalar Multiple
Given a vector v and a real number (called a scalar), we define the scalar multiple of αv of v by:
(1) α>0, αv has magnitude α|v| and same direction as v
(2) α<0, αv has magnitude |α||v| and direction of -v
(3) α=0, αv = 0
position vector
Given a point A in three dimensions, with origin O nominated. Then the position vector a of A is OA
unit vector
A non-zero vector v is a unit vector if |v|=1
parallel vectors
two vectors which either have
(i) same direction
(ii) opposite direction
unit vectors parallel to non-zero vector
v/|v| and -|v|/v
scalar product
Given tow vectors u and v, the scalar product is denoted by u.v and if defined as follows:
(i) u!+0,v!=0. Then
u.v = |u||v|cosθ
Where θ is the non-reflex angle between u and v
(ii) u=0 or v =0
Then we have u.v=0
vector projection
Given two vectors u and v, the projection of v unto u is given by
projᵤv = (v.u/|u| ) u/|u|
vector product
The vector product of two vectors u and v in E³ is denoted by uxv. The definition is split into three cases:
(i) u,v!=0 and u not parallel to v
(a) the magnitude of |uxv|= |u||v|sinθ
(b) the direction of uxv is perpendicular to both u and v
(ii) u and v ae non-zero parallel vectors uxv=0
(iii) if u=0 or v=0 then uxv=0
Scalar triple product
The scalar triple product of three vectors u, v, w is u(vxw) (volume of a parallelipiped)
Parabola
A parabola is the set of all points in the plane that are equidistant from a given fixed point and a fixed line in the plane
Ellipse
An ellipse is the set of points in a plane whose distances from two fixed points in the plane have a constant sum
Eccentricity of an ellipse
The eccentricity of an ellipse is given by e = c/a = √(a²-b²)/a
directrices of an ellipse
For an ellipse with foci F₁(c,0) and F₂(-c,0) and a standard equation. Then the lines with equations c =±a/e are called the directrices of the ellipse
Hyperbola
A hyperbola is the set of points in a plane whose distances from two fixed points in the plane have a constant difference.
Eccentricity of a hyperbola
The eccentricity of a hyperbola is given by e = c/a = √(a²-b²)/a
Directrices of hyperbola
For an hyperbola with foci F₁(c,0) and F₂(-c,0) where c>0, and a standard equation. Then the lines with equations c =±a/e are called the directrices of the ellipse
Rectangular Hyperbola
Consider a hyperbola with asymtotes parallel to the x and y axis. This is known as a rectangular hyperbola, with a standard equation xy=c
