Applied Mathematics/The Basics
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The Basics of linear algebra[edit]
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 ith row and the jth 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 mbyn matrix, and m and n are called its dimensions.
Basic operation^{[1]}[edit]
Operation  Definition  Example 

Addition  The sum A+B of two mbyn 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 mbyn matrix A is the nbym matrix A^{T} (also denoted A^{tr} or ^{t}A) formed by turning rows into columns and vice versa:

Practice problems[edit]
(1)
(2)
(3)
Matrix multiplication[edit]
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 mbyn matrix and B is an nbyp matrix, then their matrix product AB is the mbyp matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B^{[2]}
 ^{[3]}
Example[edit]
Practice Problems[edit]
(1)
(2)
Dot product[edit]
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;
or
Suppose and The dot product is
Example[edit]
Suppose and
Practice problems[edit]
(1) and
(2) and
Cross product[edit]
Cross product is defined as follows:
Or, using detriment,
where is unit vector.
Reference[edit]
 ↑ Sourced from Matrix (mathematics), Wikipedia, 28th March 2013.
 ↑ Sourced from Matrix (mathematics), Wikipedia, 30th March 2013.
 ↑ Sourced from Matrix (mathematics), Wikipedia, 30th March 2013.