1. Definitions and Operations

Definitions

Vector. Let be a positive integer and let denote the set of real numbers, then is the set of all -tuples of real numbers. A vector is an -tuple of real numbers. The notation "" is read "element of ". For example, consider a vector that has three components:

Matrix. A matrix is a rectangular array of real numbers with rows and columns. For example, a matrix looks like this:

Vector Operations

We now define the math operations for vectors. The operations we can perform on vectors and are:

Addition
Subtraction
Scaling
Dot product
Cross product
Norm length

The dot product and the cross product of two vectors can also be described in terms of the angle between the two vectors. The formula for the dot product of the vectors is . We say two vectors and are orthogonal if the angle between them is .
The dot product of orthogonal vectors is zero: .
The norm of the cross product is given by .
The cross product is not commutative: , in fact

Matrix Operations

We denote by the matrix as a whole and refer to its entries as . The mathematical operations defined for matrices are the following:

Addition	Subtraction is the inverse of addition
Transpose
Trace
Product	The product of matrices and is another matrix given by formula: Note that the matrix product is not a commutative operation: For Matrix-Vector Product, see here Matrix-Vector Product
Inverse	By definition, the inverse matrix undoes the effects of the matrix . The cumulative effect of applying after is the identity matrix More about Computing the Inverse
Determinant	The determinant of a matrix, denoted or , is a special way to combine the entries of a matrix that serves to check if a matrix is invertible or not. The determinant formulas for and matrices are , and If the then is not invertible. If then is invertible.

Additional Topics

Using Matrices for Solving Systems of Equations: Systems of Equations

Suppose we're asked to solve the following system of equations:

Without a knowledge of linear algebra, we could use substitution, elimination, or subtraction to find the values of the two unknowns and .

Gauss-Jordan elimination is a systematic procedure for solving systems of equations based the following row operations:

Adding a multiple of one row to another row

Swapping two rows

Multiplying a row by a constant

Basis, Dimensions, Eigenvalues and eigenvectors etc: extra

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