Page 29 - Computational Colour Science Using MATLAB

16 LINEAR ALGEBRA FOR BEGINNERS

2.3 Transposes and inverses

T
If A is an m n matrix, then the transpose of A, denoted by A , is defined to be
the n m matrix that results from interchanging the rows and columns of A; that
T
T
is, the first column of A is the first row of A, the second column of A is the
second row of A, and so forth.
If A is a square matrix and a matrix A 1 can be found such that
1
AA 1 ¼ A A ¼ I,
where I is the identity matrix, then A is said to be invertible and A 1 is the inverse
of matrix A.

2.4 Linear and non-linear transforms

A linear transform is a type of function; a rule f that associates with each element
in a set A one and only one element in a set B (Anton, 1994). If f associates the
element b with the element a, then we write b ¼ f(a). For the most common
functions, A and B are sets of real numbers, in which case f is a real-valued
function of a real variable <. A function may associate a four-dimensional real
4
3
value < with a three-dimensional real value < , in which case we say that f is a
4
3
3
4
transformation from < to < , or that f maps < into < . We denote this by
3
4
writing f: < !< .
The simultaneous equations
w 1 ¼ 2x 1 3x 2 þ x 3 5x 4 ,
w 2 ¼ 4x 1 þ x 2 2x 3 þ x 4 ,
w 3 ¼ 5x 1 x 2 þ 4x 3 ,
4
3
define an example of a function f: < !< .
There are no squared or higher terms in this example and therefore we can
4
3
further say that it is a linear transform T: < !< . In matrix form this example
can be expressed as follows:
2 3
3 x 1
2 3 2
w 1 2 3 1 5
6 x 2 7
4 1 2 1 ,
w 2 ¼
4 5 4 56 7
4 x 3 5
5 1 4 0
w 3
x 4
or more efficiently as
w ¼ Ax, ð2:3Þ
where w and x are 3 1 and 4 1 column matrices, respectively, and A is a 3 4
matrix. The matrix A is called the standard matrix for the linear transformation.