  Our Linear Algebra Tutorials: at a glance Linear Algebra: Eigenvalues, Eigenvectors, Cayley-Hamilton Theorem Linear Algebra - Eigenvalues, Eigenvectors and Cayley Hamilton Theorem - Outline of Contents:
Here's a quick outline of topics to be covered in this tutorial: RankIf all the minors of the matrix A of order r+1 are zero and at least one minor of order r≠0 then matrix A is said to be of rank ‘r’ Echelon form of MatrixA matrix ’A’ is said to be in echelon form if, 1. All the zero rows of ‘A’ followed by non-zero row 2. The number of zero’s before the first non-zero element in a non-zero row is less than the number of such zeros in the next row 3. The first non-zero element in a non-zero row is ‘1’ Elementary Transformations or Elementary operations of a matrix -and what they result in:The following three operations applied on the rows (columns) of a matrix are called elementary row (column) transformations. We'll take a look at what happens after making these elementary transformations.1. Interchange of any two rows(columns) 2. Multiplying all elements of a row(column) of a matrix by a non-zero scalar 3. Adding to the elements of a row (column) ,the corresponding elements of any other row(column) multiplied by any scalar k. A major focus of this tutorial will be to introduce Eigenvalues and Eigenvectors
We'll also introduce important theorems such as :Rank of the matrix ‘A’ is the number of non-zero rows in its echelon formThe product of the Eigen values of a square matrix A is |A| Cayley Hamilton Theorem: Every square matrix of order n satisfies its own characteristic equation ... and so on Complete Tutorial :
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| 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. | |
