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An array of n real numbers is called a vector and it is written

( prime operation =transposing a column to a row)

Basic vector operations
Multiplication with a constant c

Addition of x, y ∈ Rn

Inner Product or Scalar Product

Let x, y ∈ R2 and denotes the angle between the vector and the x axis. Then

In general for x, y ∈ Rn

Remark: cos θ= 0 x and y are perpendicular.
Length of a vector x ∈ Rn:

How multiplication with a constant c changes the length ?



is a vector with unit length and with direction of x.

– If then and have the same direction but different length .

Unit vectors in R2:

Unit vectors in Rn:


Definition: The space of all n-tuples with scalar multiplication and addition as defined above,
is called a vector space.

Definition: is a linear combination of the vectors .
The zero vector is defined as

Definition: The vectors  are said to be linearly dependent if there exist k numbers
not all zero , such that

Otherwise is said to be linearly independent .



and are linearly independent , because

if then

(ii) Similarly you can prove in Rn that are linearly independent .

(iii) Let

Then and are linearly dependent since

Definition: A set of m linearly independent vectors in Rm called a basis for the vector space
of m-tuples.

Theorem: Every vector in Rm can be expressed as a unique linear combination of a fixed

Example Let be a basis in Rm. Then

Definition: The length of a vector x is

Definition: The inner product or dot product of two vectors x, y ∈ Rm is


(i) Length of a vector x:

(ii) Let us denote θ the angle between two vectors x, y ∈ Rm. Then

Definition: When the angle between two vectors x and y is θ = 90°or 270°we say that x
and y are perpendicular or orthogonal

Since cos 90°= cos 270°= 0 x and y are perpendicular if x'y = 0.



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