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 nth power of a matrix A is the nth power of the eigenvalue of A 8. More properties: Sum of eigenvalues of a 2x2 matrix equals the sum of elements in the principal diagonal; product of eigenvalues equals the determinant 9. Computing the eigenvalues and eigenvectors 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 eigenvalue method to get a formula for the nth 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, Nilpotent, Singular, NonSingular, Unitary matrices.
 Introduction to Matrices  Part II Problems and solved examples based on the subtopics 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 cofactors. 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 nonhomogeneous 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 subspace, demonstrating that something is not a subspace 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, GrahamSchmidt process, coordinate 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.


