Applied Mathematics/The Basics
The Basics of linear algebra
A matrix is composed of a rectangular array of numbers arranged in rows and columns. The horizontal lines are called rows and the vertical lines are called columns. The individual items in a matrix are called elements. The element in the i-th row and the j-th column of a matrix is referred to as the i,j, (i,j), or (i,j)th element of the matrix. To specify the size of a matrix, a matrix with m rows and n columns is called an m-by-n matrix, and m and n are called its dimensions.
|Addition||The sum A+B of two m-by-n matrices A and B is calculated entrywise:
|Scalar multiplication||The scalar multiplication cA of a matrix A and a number c (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of A by c:
|Transpose||The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa:
Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B
A row vector is a 1 × m matrix, while a column vector is a m × 1 matrix.
Suppose A is row vector and B is column vector, then the dot product is defined as follows;
Suppose and The dot product is
Cross product is defined as follows:
Or, using detriment,
where is unit vector.