**An Example of
Unification from**

**The Keys to Linear
Algebra**

1.3.3 Unifying the Equations of a Plane

In Section 1.3.2., you were introduced to the
following two ways to represent a plane in *n*-space:

z=m_{1}x_{1}+^{... }+m_{n-1}x_{n-1}+ b (1.32)

a_{1}(x_{1}-x'_{1}) +^{ ... }+a_{n}(x_{n}-x'_{n}) = 0 (1.33)

It would be convenient if there were one way to
represent the equation of a plane that includes both (1.32) and
(1.33) as special cases. The process of creating this new,
encompassing representation is an example of another mathematical
technique called **unification**. This technique
involves combining two or more concepts (problems, theories, and
so on) into a single framework from which you can study each of
the special cases. To illustrate, unification is used now to
combine the special cases in (1.32) and (1.33) into a single
equation for representing a plane in *n*-space.

Identifying Similarities and Differences

The first step in unification is to identify similarities and differences between the special cases in an attempt to find the commonalities. For instance, (1.32) and (1.33) exhibit the following similarities and differences.

**Similarities in (1.32) and (1.33)**

1. Both equations involve *n* variables:
*z* and *x*_{1}, ..., *x*_{n-1, }in
(1.32) and *x*_{1}, ..., *x*_{n},
in (1.34).

2. Each of the variables in both equations is
multiplied by a known constant.For example, in (1.32), *x*_{1}
is multiplied by *m*_{1}and in (1.33), *x*_{1}
is multiplied by *a*_{1}.

3. Both equations involve other known
constants: *b*, in (1.32) and *a*_{1} *x*'_{1},
..., *a*_{n} *x*'_{n}, in (1.33).

**Differences in (1.32) and (1.33)**

1. The left and right sides of the two
equations have different forms. For example, the right side of
(1.32) contains variables together with the constant *b*.
The right side of (1.33) contains only the constant 0.

2. The specific constants that multiply the variables are different in the two equations.

3. One of the variable names in (1.32) differs
from that in (1.33). Specifically, (1.32) contains the variable *z*
and (1.33) contains *x*_{n}.

The objective now is to find ways to eliminate the differences so that you can unify (1.32) and (1.33) into a single equation.

Eliminating the Differences

You can eliminate the first difference by rewriting (1.32) and (1.33) so that all terms involving variables are on the left side of the equality signs and all terms involving constants are on the right side. For instance, moving all terms involving variables in (1.32) to the left side results in

-m_{1}x_{1}-^{ ... }-m_{n-1}x_{n-1}+z = b(1.34)

Likewise, removing the parentheses in (1.33) and moving all constant terms to the right side results in

a_{1}x_{1}+^{...}+a_{n}x_{n}=a_{1}x'_{1}+ ... +a_{n}x'_{n}(1.35)

Now, in both (1.34) and (1.35), all variables are on the left side of the equality sign and only known constants are on the right side.

The second difference identified above is overcome by introducing new notation, as done, for example, in the following unification of (1.34) and (1.35):

c_{1}y_{1}+^{...}. + c_{n}y_{n }= d(unification) (1.36)

In the unification in (1.36), the new symbols *y*_{1},
..., *y*_{n }are used to represent the *n*
variables from the special cases in (1.34) and (1.35). The
symbols *c*_{1} through *c*_{n} in
the unification represent known constants that multiply the
corresponding variables in the special cases. The symbol *d*
in the unification represents a known value on the right side of
the equalities in the special cases.

As you have seen, the use of appropriate
notation is important. To avoid confusion, all symbols used in
the unified equation (1.36) are different from those in the
special cases. Often, however, mathematicians use *overlapping
notation*, that is, the same symbol is used more than once,
with a different meaning in each case. For example, you can write
(1.36) equally well as

a_{1}x_{1}+^{...}+a_{n}x_{n}=b(1.37)

In this case, the symbols *a*_{1},
..., *a*_{n} in (1.37) overlap with those same
symbols in (1.33) and the symbol *b* in (1.37) overlaps
with that same symbol in (1.32). Also, the symbols *x*_{1},
..., *x*_{n-1 }in (1.37) overlap with those same
symbols in (1.32) and (1.33). When this overlapping notation
arises, be sure to keep the meaning of the symbols straight in
your mind.