Page 154 - 35Linear Algebra

154 Matrices

7.5.4 Homogeneous Systems

Theorem 7.5.1. A square matrix M is invertible if and only if the homoge-
neous system
Mx = 0
has no non-zero solutions.

Proof. First, suppose that M −1 exists. Then Mx = 0 ⇒ x = M −1 0 = 0.
Thus, if M is invertible, then Mx = 0 has no non-zero solutions.
On the other hand, Mx = 0 always has the solution x = 0. If no other
solutions exist, then M can be put into reduced row echelon form with every
variable a pivot. In this case, M −1 can be computed using the process in the
previous section.

7.5.5 Bit Matrices

In computer science, information is recorded using binary strings of data.
For example, the following string contains an English word:

011011000110100101101110011001010110000101110010

A bit is the basic unit of information, keeping track of a single one or zero.
Computers can add and multiply individual bits very quickly.
In chapter 5, section 5.2 it is explained how to formulate vector spaces over
fields other than real numbers. In particular, al of the properties of a vector
space make sense with numbers Z 2 = {0, 1} with addition and multiplication
given by the following tables.

+ 0 1 × 0 1
0 0 1 0 0 0
1 1 0 1 0 1

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