# Vector Space

**DEFINITION1****:** Let be a set with at least two distinct elements, and be two functions. If the following conditions hold, then is called a field:

**F1)** for all ,

**F2)** for all ,

**F3)** there exists such that for all ,

**F4)** for each , there exists such that ,

**F5)** for all ,

**F6)** for all ,

**F7)** for all ,

**F8)** there exists such that for all ,

**F9)** for all , there exists such that .

**EXAMPLE1:** Each of the sets Rational Numbers, Real Numbers and Complex Numbers is a field with usual operations of addition and multiplication.

**EXAMPLE2:** Let be a prime number, , and . For , we define and . Then, is a field with elements.

**EXAMPLE3:** The set of Integers satisfies the conditions F1 to F8. However, but there is not an integer satisfying the equality . So, it does not satisfy F9 i.e., it is not a field.

**DEFINITION2****:** Let be a field, be a set, and be two functions. If the following conditions hold, then is called a vector space or linear space:

**L1)** for all ,

**L2)** for all ,

**L3)** there exists such that for all ,

**L4)** for each , there exists such that ,

**L5)** for all and ,

**L6)** for all and ,

**L7)** for all and ,

**L8)** for all .

Instead of the expression " is a vector space", we can say " is a -vector space". Each element of is called a vector and each element of is called a scalar. If , then is called a real vector space, and if , then is called a complex vector space. The element in the condition L3 is called zero vector. Zero vector is denoted by and the zero of the field is denoted by . A vector space is never empty because . In the condition L4, the vector satisfying the condition for each vector is called the additive inverse of .

**EXAMPLE4:** is a -vector space. Indeed, every field is a vector space over itself.

**EXAMPLE5:** We define , where . Vector addition and scalar multiplication are defined as follows:

,

for all

and

,

for all and .

Then, is an -vector space. In general, is a -vector space when is a field.

**EXAMPLE6:** If are two fields, , and , then is a -vector space. Accordingly, is an -vector space, is a -vector space and is a -vector space.

**EXAMPLE7:** If are two fields and , then is not an -vector space.

**SOLUTION:** In order that is an -vector space, the scalar product () must be defined from to . We will show that this fails. implies that there exists such that . Since , then . By , we get that must be in the field , i.e., . This is a contradiction. Consequently, is not an -vector space.

By Example7, is not an -vector space and is not a -vector space.

**EXAMPLE8:** Let be a set, be a field and

.

Vector addition and scalar multiplication are defined as follows:

For all , for each .

For all and , for each .

Then, is a -vector space and denoted by .

Consider and , then we have .

Similarly, if and , we have i.e., is the space of all real sequences and denoted by . We can write . If we take instead of , we obtain .

**EXAMPLE9:** Let be a field and be the set of all polynomials with coefficients in i.e.,

.

Vector addition and scalar multiplication are defined as follows:

Let and the polynomials and be in . Assume that the relations , and hold.

.

.

Then, is a -vector space.

**PROPOSITION1:** Let be a field, be a -vector space. Then,

**a)** The zero element is unique.

**b)** The inverse element of each vector is unique (the unique element satisfying the equality for fixed vector is denoted by ).

**c)** for each .

**d)** for each .

**e)** for each .

**f)** .

**DEFINITION3:** Let be a field, be a -vector space, and . The difference of the vectors and is denoted by and defined as

,

where is the additive inverse of the vector .