**A Reconsideration of the Vector Space ****Â**^{n}** (some geometric interpretation) **

The term **vector** is used by scientists to indicate a quantity (such as
velocity or force) that has both magnitude and direction.

A **2-dimensional position vector **is an ordered pair of real numbers. Geometrically, we mean the
directed line segment from the origin to the point in the Cartesian plane. This generalizes to **3-dimensional position vectors**.

An *n*-dimensional position vector
is an ordered sequence of real numbers. Geometrically, an *n*-dimensional position vector **v** is the directed line segment from the
origin to the point in an *n*-dimensional coordinate system. The various
are called the **components **of the vector **v**.
The set of all *n*-dimensional position
vectors is denoted .
(Note that can also be interpreted as a set of points in *n*-space.)

What are some examples of
elements of Â^{4}? Of Â^{5}?

**Special Canonical Vectors in ****Â**^{n}

The vector is the vector with all components equal to
zero except the *i*th, which is equal
to one.

## Vector Operations

** **

** **

Two vectors **v** and **w** are equal if all of their corresponding components are equal
(i.e. if each ).

Let and be two vectors in Â^{n} and let *c*
be a scalar. Then we define the operations of vector addition and scalar
multiplication as follows:

Note that either of these
operations results in another element of Â^{n}. We say that Â^{n} is **closed **under
these operations, and call it a **vector
space**.

What does vector addition and
scalar multiplication “look like” in 2-space? Investigate with and .

The **length **or **magnitude** or **norm** of a vector can always be given by the formula

Find a unit vector in the
direction of .

We can also define such terms
as **zero vector**, **negative vectors**, and **subtraction
of vectors**.

**Properties of the Vector Space ****Â**^{n}

Â^{n}
satisfies all axioms as given in the definition of an arbitrary vector space.

One use of the properties is
that they allow us to combine vectors more fluidly. For example, let ,
,
and .
Determine .

Two forces, represented by
the vectors and ,
act on a body at the origin of a 2-dimensional coordinate system. Find the
resultant force. How can we keep the force but double the magnitude of the resultant
force?

Consider the homogeneous
system with the following matrix of coefficients.

What does the solution set
look like? Given one or two particular solutions, can you generate some more?