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Basic Concepts In Linear Algebra and Vector Spaces - A Tutorial with Examples and Solved Problems


Linear Algebra














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                                                                                                                         Linear Algebra: Vector Spaces

Basic Concepts In Linear Algebra and Vector Spaces - Outline of Contents:

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. Showing how a given set is NOT a subspace over the field of real numbers R.


Summarized Recap of the above tutorial document:


Consider a set F of elements a,b,c..Etc. Suppose that the operations of addition and multiplication are defined i.e. we can find + and . for any two elements then the set F is
called a field if the following properties are satisfied. F={a ,b , c.........,+,.,......}
Closure Law: a+b and a.b are elements of F
Commutative Law: a+b = b+a and a.b = b.a
Associative law: a+(b+c)=(a+b)+c and a.(b.c)=(a.b).c
Distributive law: a.(b+c)=a.b+b.c
Additive and Multiplicative Identity: There exists an element ‘0’ and ‘1’ in F such that a+0=a and a.1=a
Additive Inverse: For every element ‘ a’ there exists an element ‘-a ’ in F such that a+(-a )=0
Multiplicative  Inverse: For every element ‘ a’ there exists an element ‘a-1 ’ in F such that aa-1=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 c1, c2, c3 such that c1x1 + c2x2 + c3x3 ... = 0 )
(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


You might like to take a look at some of our other Linear Algebra tutorials :


 Introduction to Matrices - Part I   Introduction to Matrices. Theory, definitions. What a Matrix is, order of a matrix, equality of matrices, different kind of matrices: row matrix, column matrix, square matrix, diagonal, identity and triangular matrices. Definitions of Trace, Minor, Cofactors, Adjoint, Inverse, Transpose of a matrix. Addition, subtraction, scalar multiplication, multiplication of matrices. Defining special types of matrices like Symmetric, Skew Symmetric, Idempotent, Involuntary, Nil-potent, Singular, Non-Singular, Unitary matrices.


Introduction to Matrices - Part II Problems and solved examples based on the sub-topics mentioned above. Some of the problems in this part demonstrate finding the rank, inverse or characteristic equations of matrices. Representing real life problems in matrix form.



Determinants Introduction to determinants. Second and third order determinants, minors and co-factors. Properties of determinants and how it remains altered or unaltered based on simple transformations is matrices. Expanding the determinant. Solved problems related to determinants.  Simultaneous linear equations in multiple variablesRepresenting a system of linear equations in multiple variables in matrix form. Using determinants to solve these systems of equations. Meaning of consistent, homogeneous and non-homogeneous systems of equations. Theorems relating to consistency of systems of equations. Application of Cramer rule. Solved problems demonstrating how to solve linear equations using matrix and determinant related methods. 





Basic concepts in Linear Algebra and Vector spacesTheory and definitions. Closure, commutative, associative, distributive laws. Defining Vector space, subspaces, linear dependence, dimension and bias. A few introductory problems proving certain sets to be vector spaces. Introductory problems related to Vector Spaces - Problems demonstrating the concepts introduced in the previous tutorial. Checking or proving something to be a sub-space, demonstrating that something is not a sub-space of something else, verifying linear independence; problems relating to dimension and basis; inverting matrices and echelon matrices.




More concepts related to Vector SpacesDefining and explaining the norm of a vector, inner product, Graham-Schmidt process, co-ordinate vectors, linear transformation and its kernel. Introductory problems related to these.



Problems related to linear transformation, linear maps and operators - Solved examples and problems related to linear transformation, linear maps and operators and other concepts discussed theoretically in the previous tutorial. 
Definitions of Rank, Eigen Values, Eigen Vectors, Cayley Hamilton Theorem 
Eigenvalues, eigenvectors, Cayley Hamilton Theorem




More Problems related to Simultaneous Equations; problems related to eigenvalues and eigenvectors  Demonstrating the Crammer rule, using eigenvalue methods to solve vector space problems, verifying Cayley Hamilton Theorem, advanced problems related to systems of equations. Solving a system of differential equations . 





 A few closing problems in Linear AlgebraSolving a recurrence relation, some more of system of equations.