It’s Professor Dave, let’s check out some spaces.
We’ve talked a lot about vectors and matrices lately, as well as how certain mathematical
operations like addition and scalar multiplication work on them. Now we want to generalize these
operations to define what is known as a vector space, which is sometimes also called a linear
space, so let’s look at this concept now. Before defining anything, let’s just introduce
some notation. Let’s call our vector space V. If some vector is a member, or “element”
of V, we can denote it with this symbol here, so the following means that A is an element
of V. All elements of the vector space have the same properties we already discussed for
vectors, so elements must abide by the commutative and associative properties of addition, as
is shown here. There also exists a zero vector in the vector space such that any element
plus the zero vector will just give back the element. For each element a, there exists
another element, -a, within V, such that the addition of the two gives the zero vector.
Scalars can be distributed across the addition of two members of V, just as an element can
be distributed across the addition of two scalars. The product of two scalars multiplied
by an element is the same as one of the scalars times the product of the other scalar and
the element. And finally multiplying by the scalar 1 just gives back that element. So
these are simply properties that elements of the vector space must possess. And in this
way we can regard a vector space as a collection of elements that can be added together in
any combination, and multiplied by scalars in any combination.
Now, an important property of a vector space is something called closure. Closure requires
two properties. First, for an element a of V, multiplying a by any scalar will give a
result that is also within V. Second, for any two elements a and b within V, adding
the two elements will also give a result that is contained within V. These closure properties
are what determine if V is a vector space. Let’s take some time to examine these properties.
At this point it will be useful to introduce this R symbol, which represents the set of
all real numbers, meaning positive, negative, rational, irrational, all the real numbers
we talked about when we discussed the types of numbers in a previous tutorial. We can
pick any element of this set, a real number, and multiply it by any scalar we want. Let’s
take the number 5 as our element and multiply it by a few scalars. 2 times 5 is 10. 8 thirds
times five is 40 thirds. One ninth times 5 is 5 ninths. Pi times 5 is 5 pi, and so on.
Notice that all of our results are still real numbers. This is because no matter what we
choose, a real number times a real number is still real. So the set of real numbers
satisfies the first closure property. Now for the second property, we will need to choose
two elements to add together. We could choose 74 and -10, adding them together to get 64.
Adding two and four fifths gives us fourteen fifths. No matter what two real numbers we
choose, the sum of two real numbers is still real. So because the two criteria have been
met, the set of real numbers is closed and can be considered a vector space.
Moving to another example, let’s expand on our notation. If R is the set of real numbers,
Rn is the set of real vectors with a length of some integer n. Similarly, Rm x n is the
set of real m by n matrices. First, let’s just look at R3 vectors. We can multiply any
such vector by any scalar and find that our result is a vector of length 3, made of real
numbers. Similarly we can add any two elements and find the same to be true, given that each
component is simply now a sum that will result in another real number. This set is closed,
and is thus also a vector space. Given what we know about matrix addition, if some set
is comprised of matrices with the same dimensions, multiplying by some scalar or adding any two
matrices together will not change their dimensions, which makes this, too, a vector space. So
it would seem that vector spaces can actually be made of more than just vectors. In fact,
we could define a vector space that was made up of functions if we wanted to. For example
let’s take the set of linear polynomials, in the form of ax + b. If we multiply by a
scalar c, we can distribute to get ca times x, plus bc. Since we are dealing with real
numbers, ca and bc will also be real, and the form of our equation is still a linear
polynomial, so our result is still within the set. Similarly if we have two different
equations, a1 times x plus b1, and a2 times x plus b2, and we add them, once we group
like terms we get the quantity (a1 plus a2) times x, plus the quantity (b1 plus b2). Once
again all of these are real numbers and we get another linear polynomial that is contained
in our set. With these two closure properties satisfied, we therefore have a vector space
made of functions. We’ve listed a lot of examples of vector
spaces now, so what does it look like when the closure properties are not satisfied?
Take for example a simple set made of vectors of length two. However all the elements of
this set have a value of 2 in the second row. Let’s take two elements of this set, a and
b, and try adding them together. We don’t even need to assign values for the entries
in the first row to see what will happen. The first row will be the sum of a1 and b1,
while the second row will be 2 plus 2, which is 4. Our starting set only contained elements
with the value 2 in the second row. But when we added two vectors of this set we got a
4 in the second row, which means the sum is not a member of the set. Thus the set is not
closed and is not a vector space, despite the fact that it is made of vectors. A lot
of the topics we introduce in the coming tutorials will involve working with vector spaces, so
let’s check comprehension.