From now on, we will typically write vectors as columns. Thus, when we write x ∈ Fn we typically
We will also use the following convention.
The rules for adding and multiplying by a constant remain the same. To add, you add corresponding
entries and to multiply by a scalar, you multiply every entry by the scalar. Consider the following system of
Another way to write this is
That expression on the left is called a linear combination of the three vectors listed there whenever x,y,z
are numbers. Another way to write it is
The rows of the above matrix are
The columns of this matrix
It is called a 3 ×
3 matrix because it has three rows and three columns.
More generally, we have the following definition.
Definition 5.0.1 An m×n matrix is a rectangular array of numbers which has m rows and n columns.
We write this as
Thus the entry in the ith row and the jth column is denoted as Aij. As suggested above,
Note that Ax is in Fm and the ith entry of this vector Ax is
In other words, the ith entry of Ax is the dot product of the ith row of A with the vector x.
We like to write x to denote an n × 1 matrix which is often called a vector. Then xT will denote a 1 × n
matrix or row vector.
Note that if A is m×n then Ax is an m× 1 matrix provided x is n× 1. Thus A makes a vector in Fn
into a vector in Fm.
Example 5.0.3 Show the following:
Example 5.0.4 Write the system of equations
in the form Ax = b.
According to the above, this system can be written as
The following is the most fundamental observation about multiplying a matrix times a vector.
Theorem 5.0.5 Let A be an m × n matrix and let x,y be two vectors in Fn with a,b two scalars.
Proof: By the above definition and the way we add vectors,
Since the ith entries coincide, it follows that A
as claimed. ■
Definition 5.0.6 Define some special vectors ei as follows:
Thus in F3, we would have
Observation 5.0.7 Let A be an m×n matrix. Then for ei ∈ Fn,Aei delivers the ith column of A. To see
= 0 unless j
= i when it is 1. Thus, for k arbitrary, the kth entry of Aei is Aki. Thus the
result of multiplying by ei is
which is indeed the ith column. Another way to see this is to let
the ith column of A.