What are vector spaces?
Here is a definition of a vector space:
A vector space is a set V of elements that can be added together and
multiplied by scalars.
Okay, lets see if I can make this a little clearer. The first part a set V
of elements means that it is a group of something. It can a group of
anything, but of course in math mainly it is numbers.
The second part tells us whether or not that group is a vector space. There
are two requirements:
1. The members of the group must be able to be added to each other.
2. The members of the group must be able to be multiplied by a scalar, or
a single number.
For example, if we look at the set of all real numbers, that is normal
everyday numbers,we find that this is a vector space.
1. Numbers can be added together and get a result that is a number.
i.e.: 1 + 2 = 3
2. Numbers can be multiplied by a scalar:
2 * 3 = 6
A set that would not be considered a vector space is all the odd numbers:
1. 3 + 3 = 6, which results in a number not in the set
2 2 * 3 = 6, also which is not in the set.
Those are some simple examples, and not really what vector spaces are all
about. You are probably a little familiar with the Cartesian coordinate
system. This is where we describe a point as having an x coordinat and a y
coordinate. Like on graph paper, you can number the rows and columns and
then find any point by telling how many columns it is over, and how many
rows it is up.
Points are written as two numbers: (x, y)
Some examples of points are: (1,2) 1 over, 2 up. (4,5) 4 over, 5 up.
A related concept to points is a vector. The difference between a point and
a vector is mainly in how we think about them. A point describes a
position, but a vector describes a direction.
So if we drew a line from where we started numbering on the graph paper
(0,0), to a point (1,2), this would be a vector <1,2>.
The neat thing about vectors is that they can be added together:
1. To add vectors, we add the first numbers together:
<1,2> + <4,5> = <5,7>
2. They can be multiplied by scalars, or a single number. To do this, we
multiply each part by the number:
3 * <1,2> = <3, 6>
There isn't very many other normal operations that can be done on vectors.
We can subtract them, which is similar to addition, we subtract the parts.
But they can't be multiplied together: <1,2> * <3,4> is not valid.
They can't be divided by each other: <1,2> / <3,4> is also not valid.
I hope this helps define a vector space for you.
Basically, a vector space is the set of all vectors that can be created by
linear combinations of a given set of vectors. If you take a vector and
multiply it by any real number, and take another vector and multiply it by
any real number, and then add them together, this new vector is a linear
combination of the first two. So a vector space is all the possible linear
combinations of the set of basis vectors. The basis vectors are said to
"span" the vector space. You can find different sets of basis vectors that
span the same vector space.
Richard Barrans Jr., Ph.D.
Chemical Separations Group
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Update: June 2012