LINEAR ALGEBRA

This contains:

1. Vector and Spaces

2.Matrix Transformations

3. Alternate coordinate systems

So what are vectors?

A vector space is a set of objects called vectors, which may be added together and multiplied by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field.

Applications

Some applications of the Vector spaces:
1) It is easy to highlight the need for linear algebra for physicists - Quantum Mechanics is entirely based on it. Also important for time domain (state space) control theory and stresses in materials using tensors.

2) In circuit theory, matrices are used to solve for current or voltage. In electromagnetic field theory which is a fundamental course for communication engineering, the conception of divergence, curl are important.
For other fields of engineering, computer memory extensively uses the conception of partition of matrices. If the matrices size gets larger than the space of computer memory it divides the matrices into submatrices and does the calculation.

3) Linear operator plays a key role in computer graphics. Many CAD software generates drawing using linear operators, And don't forget about cryptography.
4) Matrices can be cleverly used in cryptography. Exchanging secret information using a matrix is very robust and easy in one sense. How about MATLAB? This software is widely used in engineering fields and MATLAB's default data type is matrix.

And, of course, Linear Algebra is the underlying theory for all the linear differential equations. In the electrical engineering field, vector spaces and matrix algebra come up often.

5) Least square estimation has a nice subspace interpretation. Many linear algebra texts show this. This kind of estimation is
used a lot in digital filter design, tracking (Kalman filters), control systems, etc.

What is the difference between Vector and Space?

A vector is a member of a vector space. A vector space is a set of objects which can be multiplied by regular numbers and added together via some rules called the vector space axioms.