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 a vector.
2. Differentiate between a vector and a 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:
