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##  Our Linear Algebra Tutorials: at a glanceLinear Algebra - Matrices Part I - A Tutorial with ExamplesLinear Algerba - Matrices Part II - Tutorial with Problems and SolutionsLinear Algebra - Determinants - A Tutorial with Problems and SolutionsLinear Algebra - Simultaneous Equations in Multiple Variables Basic Concepts In Linear Algebra and Vector Spaces - A Tutorial with Examples and Solved ProblemsLinear Algebra - Introductory Problems Related to Vector SpacesLinear Algebra - More about Vector SpacesLinear Algebra - Linear Transformations, Operators and MapsLinear Algebra - Eigenvalues, Eigenvector,Cayley Hamilton TheoremLinear Algebra - Problems on Simultaneous Equations, EigenvectorsLinear Algebra - A few closing problems in Recurrence Relations                                                                                                                                                           ------------xxxx------------

Linear Algebra: Vector Spaces

 Theory and definitions. Closure, commutative, associative, distributive laws. Defining Vector space, subspaces, linear dependence of vectors, dimension and bias. A few introductory problems proving certain sets to be vector spaces.

Just to give you an idea, here are some of the problems solved in this tutorial :

1. Problem: Show that the set of complex numbers is a vector space over the field of real
numbers.
2. Problem: Let R be a set of all real numbers and V be set of all real functions. Show that V is a vector space over R
3. Problem: Show that the set of all real polynomials of degree ‘n’ or less is a vector space over the field of real numbers.
4. Verifying whether a given set with specific operations is a vector space with vector field of real numbers.
5. subspace over the field of real numbers R.

Consider a set F a,b,c.multiplication are defined i.e. we can find + and . for any two elements then the set F
called a field if the following properties are satisfied. F={a ,b , c.........,+,.,......}
Closure Law: a+b a.b F
Commutative Law: a+b = b+a a.b = b.a
Associative law: a+(b+c)=(a+b)+c a.(b.c)=(a.b).c
Distributive law: a.(b+c)=a.b+b.c
Additive Inverse: For every element ‘ a’‘-a
Multiplicative  Inverse: For every element ‘ a’‘a-1 aa-1

We then extend these postulates to illustrate the conditions under a which a set of vectors may be called a vector space over a field.
We introduce the
(a) Cancellation Law
(b) Definition of Subspaces
(c) Condition for linear dependence of vectors. (i.e, there should be constants c123112233 ...
(d) Dimension and basis of vectors. If there are 'n' linearly dependent vectors - while every other set of ‘n+1 ’ vectors is linearly dependent then ‘ n’ is called the dimension of vector space