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.