Linear Algebra: More about Vector Spaces
Linear Algebra - More about Vector Spaces - Outline of Contents:| Defining 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. |
Here's a quick look at some of the problems we'll solve in this tutorial:
Find the co-ordinate vectors of the vector with components (2,3,4,-1) of V4 relative to the standard basis of V4. Find the co-ordinate vectors of the vector with components (2,3,4,-1) of V4 relative to the standard basis of [e1, e2, e3 ,e4] where e1 = (1,1,0,0,) e2 = (0,1,1,0) e3 = (0,0,1,1) e4 = (1,0,0,1)
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.
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