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 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.

Basic operation[1][edit]

Operation Definition Example
Addition The sum A+B of two m-by-n matrices A and B is calculated entrywise:
(A + B)i,j = Ai,j + Bi,j, where 1 ≤ im and 1 ≤ jn.

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:
(cA)i,j = c · Ai,j.
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:
(AT)i,j = Aj,i.

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 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[2]


[3]


Schematic depiction of the matrix product AB of two matrices A and B.



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]

  1. Sourced from Matrix (mathematics), Wikipedia, 28th March 2013.
  2. Sourced from Matrix (mathematics), Wikipedia, 30th March 2013.
  3. Sourced from Matrix (mathematics), Wikipedia, 30th March 2013.