**Basics**

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

as

( prime operation =transposing a column to a row)

**Basic vector operations **

Multiplication with a constant c

** Addition **of x, y ∈ R^{n}

**Inner Product or Scalar Product**

Let x, y ∈ R^{2} and
denotes the angle between the vector
and the x axis. Then

In general for x, y ∈ R^{n}

**Remark: **cos θ= 0
x and y are perpendicular.

**Length** of a vector x ∈ R^{n}:

How multiplication with a constant c changes the length ?

**Remarks:**

–Let

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

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

Unit vectors in R^{2}:

Unit vectors in R^{n}:

Let

**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 .

**Examples:**

(i)

and
are linearly independent , because

if then

(ii) Similarly you can prove in R^{n} that
are
linearly independent .

(iii) Let

Then and
are linearly dependent since

**Definition: **A set of m linearly independent vectors in R^{m} called a basis for the
vector space

of m-tuples.

**Theorem:** Every vector in R^{m} can be expressed as a unique linear combination of a
fixed

basis.

**Example** Let be a basis in R^{m}. Then

**Definition: **The length of a vector x is

**Definition: **The inner product or dot product of two
vectors x, y ∈ R^{m} is

**Remark:**

(i) Length of a vector x:

(ii) Let us denote θ the angle between two vectors x,
y ∈ R^{m}. 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.

**Notation **

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