Linear Algebra - Concepts and Examples
Generally a finite number of linear equations with a finite number of unknowns x, y, z, w, . . . is called a system of linear equations or just a linear system. Linear Algebra is the study of Linear equations Vectors and Matrices.
Euclidean space is a special case of general Vecor Spaces where the elements of $\mathbf{R}$ exists.
Euclidean space Vecors have properties like magnitude and direction. Vectors have much broader range of applications and they are generally defined as the elements of a Vector Space. These are vector spaces that constitute with matrices, functions and polynomials.
The concept of dot product is exapted to the general vector spaces as inner product. The dot product was an example of a specific operation more commonly referred to as an inner product, in a Euclidean n-space.
Tranformations maps vectors from one vector spaces to the vectors of another vector space. If for a function f we have $ f : V \rightarrow W $, where V and W are vector spaces, then the function f is called a transformation or map from V to W. Transformation is just another word for function and works on vectors as well as numbers. Transformation is denoted by $ T : V \rightarrow W $
A determinant of amatrix A is normally denoted by det(A) and is a scalar not a matrix. If B is the inverse of matrix A then $ B = A^{-1} $
EigenVector $\vec{v}$ statisfies $ T(\vec{v}) = \lambda \vec{v} $ for the transformation T, and $\lambda$ is the eigenvalue that is associated with the eigenvector $\vec{v}$. The transformation T is a linear Transformation that can be represented as $T(\vec{v}) = A\vec{v}$
Eigenvectors $\vec{v}$ are do not change direction when transformation such as T is applied to it. So if we apply T to a vector $\vec{v}$ and the result $T(\vec{v})$ is parallel to the original $\vec{v}$ then $\vec{v}$ is an eigenvector.