• ### UNIT 10 : Vector space of real numbers

Key unit competence

Determine the magnitude and angle between two vectors and to be able to plot these vectors. Also, be able to point out the dot product of two vectors.

Learning objectives

10.1  Vector spaces lR2

Definitions and operations on vectors

In Senior 2, we were introduced to vector and scalar quantities.

Task 10.1

1. Define vector.

2Differentiate between vector and scalar quantity.

3. Use diagrams to illustrate equal vectors.

A quantity which has both magnitude and direction is called a vector. It is usually represented by a directed line segment. In our daily life, we deal with two mathematical quantities:

(a) one which has a defined magnitude but for which direction has no meaning (example: length of a piece of wire).

In physics, length, mass and speed have magnitude but no direction. Such quantities are defined as scalars

(b) the other for which direction is of fundamental significance. Quantities such as force and wind velocity depend very much on the direction in which they act. These are vector quantities.

Addition of vectors

Vectors are added by the triangle law or end-on rule or parallelogram law of addition.

The parallelogram law

Vector subtraction

Multiplication of a vector by a scalar

When a vector is multiplied by a scalar (a number) its magnitude changes. If it is multiplied by a positive number the direction remains the same.

However, if it is multiplied by a negative number, the direction of the vector reverses.

10.2 Vector spaces of plane vectors ( lR, V, +)

The definition of vector space denoted by V needs the arbitrary fields F = whose elements are called scalars.

Definitions and operations

Properties of vectors

Linear combination of vectors

Spanning vectors

Linear dependent vectors

Linear independent vectors

Basis and dimension of a vector space

Dimension of vector space

The dimension of a non-zero vector space V is the number of vectors in a basis for V. Often we write Dim (V) for the dimension of V.

For lRn , one basis is the standard basis, and it has n vectors. Thus, the dimension of n is lRn.

10.3 Euclidian vector space

Activity 10.1

In groups, carry out research to determine the similarities between vector spaces and Euclidian spaces.

Dot product and properties

These are vectors:

UNIT 9 : Differentiation of polynomials, rational and irrational functions, and their applications UNIT 11 : Concepts and operations on linear transformations in 2D