Linear Algebra: Problems Based on Simultaneous Equations, Eigenvalues, Eigenvectors
Linear Algebra - Problems Based on Simultaneous Equations, Eigenvalues, Eigenvectors - Outline of Contents:
| 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 . |
We will focus on problems from a variety of topics in Linear Algebra.
1. Solving a system of linear equations using augmented matrix method and using echelon forms.
2. Finding the condition for a system of linear equations to be consistent.
3. Testing a system of linear equations for consistency.
4. Applying Cramer's rule
5. Handling a system which does not have a unique solution.
6. Finding the characteristic equation of a matrix.
7. Proving properties related to eigenvalues: such as, the eigenvalue of the n-th power of a matrix A is the n-th power of the eigen-value of A
8. More properties: Sum of eigen-values of a 2x2 matrix equals the sum of elements in the principal diagonal; product of eigen-values equals the determinant
9. Computing the eigen-values and eigen-vectors of 2x2 and 3x3 matrices.
10. Checking for the conditions when a
homogeneous system have
11. Verifying Caley Hamilton theorem for a matrix; finding inverse of a matrix.
12. Using the eigen-value method to get a formula for the n-th power of a matrix A.
13. Solving a system of linear and simultaneous differential equations.
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.|| |