VLGA definitions term 1

Question 1

Set

Answer 1

A collection of things

Question 2

order

Answer 2

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

Question 3

cartesian product

Answer 3

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

Question 4

difference (of A and B)

Answer 4

those elements which are in A but not in B

Question 5

commutative

Answer 5

order of operations doesn’t matter

Question 6

Associative

Answer 6

placement of brackets doesn’t matter

Question 7

distributive

Answer 7

expansion of brackets

Question 8

complement of A

Answer 8

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

Question 9

mathematical induction

Answer 9

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

Question 10

linear equation

Answer 10

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

e.g. ax+by=c

Question 11

solution to linear equations

Answer 11

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

Question 12

homogeneous

Answer 12

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.

Question 13

non homogeneous

Answer 13

The constants in the simultaneous equations are equal to anything

Question 14

matrices are equal

Answer 14

if

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

Question 15

matrix

Answer 15

array of numbers

Question 16

matrix order

Answer 16

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

Question 17

sub-matrix

Answer 17

a part of a bigger matrix

Question 18

zero matrix

Answer 18

a matrix where all the elements are equal to zero

Question 19

diagonal matrix

Answer 19

An nxn matrix is diagonal when:

aᵢⱼ = 0 for i!=j

Question 20

identity matrix

Answer 20

A diagonal matrix where the the diagonal is all 1s

Question 21

Upper Triangular matrix

Answer 21

any nxn matrix where aᵢⱼ = 0 where i>j

Question 22

Lower Triangular matrix

Answer 22

any nxn matrix where aᵢⱼ = 0 where j>i

Question 23

inverse matrix

Answer 23

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

Question 24

singular

Answer 24

the matrix doesn’t have an inverse

Question 25

Elementary row operation

Answer 25

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

Question 26

row echelon form

Answer 26

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

Question 27

Guassian Elimination

Answer 27

applying successive EROs to get the matrix into echelon form.

Question 28

Elementary matrix

Answer 28

An nxn elementary matrix is obtained by performing one elementary row operation to the identity matrix I

Question 29

vector quantity

Answer 29

a quantity has both magnitude and direction

Question 30

scalar quantity

Answer 30

a quantity that has only magnitude

Question 31

set of vectors in three dimensions

Answer 31

all three vectors together with the zero vector which has no magnitude or direction.

Question 32

vector alternative

Answer 32

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

Question 33

vector equality

Answer 33

Two vectors are equal if and only if they have the same magnitude and direction

Question 34

Vector Addition

Answer 34

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

Question 35

Vector Subtraction

Answer 35

Vector subtraction is defined as follows:

u-v = u+(-v)

Question 36

Scalar Multiple

Answer 36

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

Question 37

position vector

Answer 37

Given a point A in three dimensions, with origin O nominated. Then the position vector a of A is OA

Question 38

unit vector

Answer 38

A non-zero vector v is a unit vector if |v|=1

Question 39

parallel vectors

Answer 39

two vectors which either have

(i) same direction
(ii) opposite direction

Question 40

unit vectors parallel to non-zero vector

Answer 40

v/|v| and -|v|/v

Question 41

scalar product

Answer 41

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

Question 42

vector projection

Answer 42

Given two vectors u and v, the projection of v unto u is given by
projᵤv = (v.u/|u| ) u/|u|

Question 43

vector product

Answer 43

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

Question 44

Scalar triple product

Answer 44

The scalar triple product of three vectors u, v, w is u(vxw) (volume of a parallelipiped)

Question 45

Parabola

Answer 45

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

Question 46

Ellipse

Answer 46

An ellipse is the set of points in a plane whose distances from two fixed points in the plane have a constant sum

Question 47

Eccentricity of an ellipse

Answer 47

The eccentricity of an ellipse is given by e = c/a = √(a²-b²)/a

Question 48

directrices of an ellipse

Answer 48

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

Question 49

Hyperbola

Answer 49

A hyperbola is the set of points in a plane whose distances from two fixed points in the plane have a constant difference.

Question 50

Eccentricity of a hyperbola

Answer 50

The eccentricity of a hyperbola is given by e = c/a = √(a²-b²)/a

Question 51

Directrices of hyperbola

Answer 51

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

Question 52

Rectangular Hyperbola

Answer 52

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