Obviously, we could do something similar for any system of linear equations. Thus, a
simpler way to write Definition 18.1.19 is as follows. A linear system of equations having
m equations and n variables is of the form
( x )
( ) | 1. |
Ax ≡ a1 ⋅⋅⋅ an ( .. ) ≡ x1a1 + ⋅⋅⋅+ xnan = b (18.9)
xn
(18.9)
where A is the matrix having the columns a_{i} given above. It is an m×n matrix because
it has m rows and n columns. The right side b is m× 1. The idea is that for x a vector,
Ax is another vector and you want to choose x in such a way that Ax = b. The matrix A
has an entry A_{ij} which is in the i^{th} row and j^{th} column. Then another way to write 18.9
is
( ) ( ) ( ) ( )
a11 a12 a1n b1
x1 |( ... |) + x2 |( ... |) + ⋅⋅⋅+ xn|( ... |) = |( ... |)
a a a b
m1 m2 mn m
or more simply,
b = ∑ A x
i j ij j
A matrixis a rectangular array of numbers as in 18.9. In this book, they will
be real numbers. However, nothing is changed if they are complex numbers.
To emphasize this fact, I will sometimes write F to indicate the name of the
numbers used. Thus F could be either ℝ or the complex numbers ℂ. Several of
these rectangular arrays are referred to as matrices. For example, here is a
matrix.
( 1 2 3 4)
( 5 2 8 7)
6 − 9 1 2
The size or dimension of a matrix is defined as m×n where m is the number of rows and
n is the number of columns. The above matrix is a 3 × 4 matrix because there are three
rows and four columns. The first row is
(1234)
, the second row is
(5287)
and so
forth. The first column is
( )
1 5 6
^{T}. When specifying the size of a matrix,
you always list the number of rows before the number of columns. Also, you
can remember the columns are like columns in a Greek temple. They stand
upright while the rows just lie there like rows made by a tractor in a plowed
field. Elements of the matrix are identified according to position in the matrix.
For example, 8 is in position 2,3 because it is in the second row and the third
column. You might remember that you always list the rows before the columns
by using the phrase Rowman Catholic. The symbol,
(aij)
refers to a matrix.
The entry in the i^{th} row and the j^{th} column of this matrix is denoted by a_{ij}
or A_{ij}. Using this notation on the above matrix, a_{23} = 8,a_{32} = −9,a_{12} = 2,
etc.
You think of a matrix as an explicit way to compute a linear transformation. Let
A =
(a1 ⋅⋅⋅ an )
and let x,y be two n× 1 matrices, vectors in F^{n}. Then if a,b are
numbers (called scalars in this context),
A (ax+ yb)
is going to be an m × 1 vector. Its i^{th} component is by definition
This is what it means to say that matrix multiplication is an example of a linear
transformation. Here is the formal definition of a linear transformation.
Definition 18.2.1A function T : F^{n}→ F^{m}is called a lineartransformationif for any scalars a,b and vectorsx,yin F^{n},T
(ax + by )
=
aTx + bTy.
The important observation about this is the following theorem. First is some standard
notation.
Definition 18.2.2Define e_{i}as the column vector with a 1 in the i^{th}positionfrom the top and a zero everywhere else. The size of e_{i}will be determined by context.Thus
( )T ( )T
e1 ≡ 1 0 ⋅⋅⋅ 0 ,e2 ≡ 0 1 ⋅⋅⋅ 0 ,etc.
Now with the above definition, here is an important observation.
Theorem 18.2.4If T : F^{n}→ F^{m}is a linear transformation, then thereexists a unique m×n matrix A such that Tx = Ax where on the right the meaningis matrix multiplication of a vector.
Proof:This follows from the following computation resulting from the fact that T is
linear.
( )
(∑ ) ∑ ( ) x1 ( )
Tx = T xiei = xiT ei = Te1 ⋅⋅⋅ T en |( ... |) = Te1 ⋅⋅⋅ T en x
i i xn
Thus there is a matrix A and it must equal
A = (T e ⋅⋅⋅ T e )■
1 n
Because of this theorem, we will regard matrices as linear transformations whenever
convenient. This is as far as I will take this because this is not a linear algebra book, but
in linear algebra, one considers the notion of a matrix of a linear transformation in terms
of an arbitrary basis. Bases will be discussed a little later. The big problem in
linear algebra is to determine whether two matrices represent the same linear
transformation.
The next section gives an example of the usefulness of this concept of representing a
linear transformation in terms of a matrix.