Linear Algebra

Today, June 1 I have a Linear Algebra final. In the spirit of the subject and trying to simultaneously study I'll talk about some subjects in the field.

I. Matrices

-Definition: A matrix is a rectangular array of entries wherein the entry can be a number or a variable.

-Uses: Matrices can be used to solve systems of equations.

-Example of a matrix:

( 1 3 4 5) -> a 1x4 matrix

-Example of solving a system:

This represents the system: x+9y=13, 20x+55y=4y using the properties of matrices, we can reduce the matrix to get x by itself and simultaneously get y by itself. x=-5.432 y=2.048

II. Determinants

-Definition: a determinant is a unique number assigned to only a square matrix. note a square matrix is a matrix with equal number of rows as columns

-Uses: Easier ways of solving systems, Finding the inverses of a matrix

-Example: from the previous matrix delete the last column to get a 2x2 matrix. the determinant of that matrix would be: -125

III. Vector Spaces

-Definition: a vector space is a set on which vector addition and scalar multiplication are defined, and certain axioms hold.

-example: a common vector space is the set of all ordered pairs of real numbers with vector operations is a vector space

IV. Inner Product Spaces

-definition:an inner product of of a vector space associates a real number with a pair of vectors

-example: a simple example that is already defined is a the dot product, which is just the components of a vector multiplies then all added up.

V. Linear Transformations

-Definition: a Linear transformation preserves the operations on a set but alter the starting position of the basis vectors.

-uses: you got me...

VI. Eigen values/vectors

-Eigen values are a large topic with a wide range of uses and nuances. to try to summarize it here would still be a large effort with more text you care for. I'll make a separate post about that later