Euclidean space geometry: vector operations
The combination of these words may sound frightening. In fact, if you want to succeed with matrix algebra, you need to start drawing inspiration from geometry as early as possible.
Sum of vectors
Definition. The set of all -dimensional vectors
with
is denoted
and is called a Euclidean space.
is a plane. The space we live in is
Our intuition doesn't work in dimensions higher than 3 but most facts we observe in real life on the plane and in the 3-dimensional space have direct analogs in higher dimensions. Keep in mind that
can be called a vector or a point in
depending on the context. When we think of it as a vector, we associate with it an arrow that starts at the origin
and ends at the point
Careful inspection shows that the sum of two vectors is found using the parallelogram rule in Figure 1. The rule itself comes from physics: if two forces are applied to a point, their resultant force is found by the parallelogram rule. Whatever works in real life is guaranteed to work in Math.

Figure 1. Sum of vectors
Exercise 1. Let be the unit vector of the
axis and
the unit vector of the
axis. Find the sum
Generalization: if
runs over the whole
axis and
runs over the whole
axis, what is the set of resulting sums
Further generalization: on the plane take two straight lines
and
that pass through the origin and are not parallel to one another. If
runs over
and
runs over
, what is the set of resulting sums
This seemingly innocuous exercise leads to profound ideas, to be considered later. The answer for the last question is that the sums will cover the whole
This fact is written like this:
Note that I intentionally use words that emphasize movement and geometry: "runs over" and "covers the whole". One of the differences between elementary and higher mathematics is that the former deals with fixed elements and the latter with sets within which there are movement and change.
Multiplication of a vector by a number
If is a number and
is a vector, we put
(scaling or multiplication by a number). Scaling
by a positive number
means lengthening it in case
and shortening in case
. Scaling by a negative number means, additionally, reverting the direction of
, see Figure 2.

Figure 2. Scaling a vector
Exercise 2. Take a nonzero vector and find the set of all products
where
runs over
The answer is that is the straight line drawn through the vector
or, alternatively, the straight line drawn through the origin and point
The expression is called a linear combination of vectors
with coefficients
To find
you first scale
and
and then add the results.
Exercise 3. Let be two nonparallel vectors on the plane. Describe the set
verbally and geometrically.
Verbally, this is the set of all linear combinations that result when
run over
Geometrically, this will be the whole plane
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