**An Example of
Translating Visual Images to Symbolic Form in**

**The Keys to Linear
Algebra**

**1.2 Arithmetic
Operations on Vectors**

In this section, various arithmetic operations
on vectors used in problem solving are described. Because a
vector has both a geometric representation (as a point or as an
arrow) and an algebraic representation (as an ordered list of
real numbers), each operation has a geometric representation with
an algebraic counterpart. The advantage of the geometric version
is that you can visualize the operation, at least for vectors in
2-space and 3-space. The algebraic version has two distinct
advantages: (1) the algebraic operation is applicable to vectors
in *n*-space, for any positive integer *n* and (2)
computers are capable of performing algebraic operations but not
geometric ones.

When an operation on a vector is described in
an algebraic (or *symbolic*) form, you should use
visualization to create an appropriate image of the operation.
Similarly, for each operation described in a geometric form, you
should create a corresponding symbolic form by using a skill
hereafter referred to as **translating visual images to
symbolic form**.

To illustrate both the technique of visualization and its counterpart, translating visual images to symbolic form, some operations are described in this section first in geometric form and then translated to symbolic form. Other operations are described first in symbolic form and then visualization is used to create an appropriate geometric image.