VLGA definitions term 1 Flashcards

1
Q

Set

A

A collection of things

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2
Q

order

A

how many elements a set has (do not count repeated elements twice)

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3
Q

cartesian product

A

the collection of ordered pairs obtained by the product of two non empty sets

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4
Q

difference (of A and B)

A

those elements which are in A but not in B

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5
Q

commutative

A

order of operations doesn’t matter

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6
Q

Associative

A

placement of brackets doesn’t matter

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7
Q

distributive

A

expansion of brackets

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8
Q

complement of A

A

All the elements in the universal set which are not in A

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9
Q

mathematical induction

A
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
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10
Q

linear equation

A

an equation that has no products of unknowns and no powers.

e.g. ax+by=c

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11
Q

solution to linear equations

A

the system has a solution whenever the values simultaneously satisfy all the equations

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12
Q

homogeneous

A

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.

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13
Q

non homogeneous

A

The constants in the simultaneous equations are equal to anything

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14
Q

matrices are equal

A

if

(i) both have the same dimension mxn
(ii) aᵢⱼ = bᵢⱼ for all i,j

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15
Q

matrix

A

array of numbers

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16
Q

matrix order

A

mxn where m is the number of rows and n is the number of columns`

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17
Q

sub-matrix

A

a part of a bigger matrix

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18
Q

zero matrix

A

a matrix where all the elements are equal to zero

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19
Q

diagonal matrix

A

An nxn matrix is diagonal when:

aᵢⱼ = 0 for i!=j

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20
Q

identity matrix

A

A diagonal matrix where the the diagonal is all 1s

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21
Q

Upper Triangular matrix

A

any nxn matrix where aᵢⱼ = 0 where i>j

22
Q

Lower Triangular matrix

A

any nxn matrix where aᵢⱼ = 0 where j>i

23
Q

inverse matrix

A

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

24
Q

singular

A

the matrix doesn’t have an inverse

25
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
26
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
27
Guassian Elimination
applying successive EROs to get the matrix into echelon form.
28
Elementary matrix
An nxn elementary matrix is obtained by performing one elementary row operation to the identity matrix I
29
vector quantity
a quantity has both magnitude and direction
30
scalar quantity
a quantity that has only magnitude
31
set of vectors in three dimensions
all three vectors together with the zero vector which has no magnitude or direction.
32
vector alternative
the subset of all line segments with the same length and direction
33
vector equality
Two vectors are equal if and only if they have the same magnitude and direction
34
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
35
Vector Subtraction
Vector subtraction is defined as follows: | u-v = u+(-v)
36
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
37
position vector
Given a point A in three dimensions, with origin O nominated. Then the position vector a of A is OA
38
unit vector
A non-zero vector v is a unit vector if |v|=1
39
parallel vectors
two vectors which either have (i) same direction (ii) opposite direction
40
unit vectors parallel to non-zero vector
v/|v| and -|v|/v
41
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
42
vector projection
Given two vectors u and v, the projection of v unto u is given by projᵤv = (v.u/|u| ) u/|u|
43
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
44
Scalar triple product
The scalar triple product of three vectors u, v, w is u(vxw) (volume of a parallelipiped)
45
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
46
Ellipse
An ellipse is the set of points in a plane whose distances from two fixed points in the plane have a constant sum
47
Eccentricity of an ellipse
The eccentricity of an ellipse is given by e = c/a = √(a²-b²)/a
48
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
49
Hyperbola
A hyperbola is the set of points in a plane whose distances from two fixed points in the plane have a constant difference.
50
Eccentricity of a hyperbola
The eccentricity of a hyperbola is given by e = c/a = √(a²-b²)/a
51
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
52
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