A Reconsideration of the Vector Space Ân (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 Â4?  Of Â5?

 

 

Special Canonical Vectors in Ân

 

 

The vector  is the vector with all components equal to zero except the ith, 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 Ân 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 Ân. We say that Ân 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 Ân

 

 

Ân 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?