Vector Spaces and Subspaces

Now that we have some basics with vectors, matrices, and linear systems, we need to build up the theoretical machinery of linear algebra that will allow us to do the cool stuff. We have about a week of theory ahead of us. It's good stuff: logical, useful, and interesting. It's also full of new definitions we need to remember and use, starting with a big generalization of what it means to be a vector.

Vector Spaces

We know vectors as lists of numbers or objects with magnitude and direction. If we look more closely, we see a long list of properties satisfied by vectors. We will list these properties. Once we do, any object that also satisfies the same properties can be treated as a vector. Then everything we know about vectors can apply to this new object. It ends up being a hugely useful abstraction! It also leads to many wonderful memes:

Let's get to that list of properties with the major definition of this section. So, what makes a vector a vector? Here are some of the things we have when we talk about vectors:

• A space containing the vectors, $\large \mathbb{R}^n$.
• Vector addition.
• Scalar multiplication.

And with these things, we have the following 10 properties (the first 5 deal with vector addition, and the second 5 deal with scalar multiplication):

1. $\large \mathbb{R}^n$ is closed under addition. In other words, the sum of two vectors in $\large \mathbb{R}^n$ is a vector in $\large \mathbb{R}^n$.
2. Vector addition is commutative: $\large \vec{v}+\vec{u}=\vec{u}+\vec{v}$.
3. Vector addition is associative: $\large \vec{u}+\left(\vec{v}+\vec{w}\right) = \left(\vec{u}+\vec{v}\right)+\vec{w}$.
4. $\large \mathbb{R}^n$ has an additive identity, a vector $\large \vec{0}$ such that $\large \vec{v}+\vec{0}=\vec{v}$.
5. Every vector $\large \vec{v}$ in $\large \mathbb{R}^n$ has an additive inverse, a vector $\large -\vec{v}$ such that $\large \vec{v}+(-\vec{v})=\vec{0}$.
6. $\large \mathbb{R}^n$ is closed under scalar multiplication. In other words, $\large c\vec{v}$ is in $\large \mathbb{R}^n$ for any $\large c$ and any $\large \vec{v}$.
7. Scalar multiplication is distributive over vector addition: $\large c\left(\vec{u}+\vec{v}\right)=c\vec{u}+c\vec{v}$.
8. Scalar multiplication is distributive over scalar addition: $\large \left(c+d\right)\vec{v}=c\vec{v}+d\vec{v}$.
9. Scalar multiplication is associative: $\large c\left(d\vec{v}\right)=(cd)\vec{v}$.
10. There is a scalar identity 1 such that $\large 1(\vec{v})=\vec{v}$.

There it is. Ten things that are properties of $\large \mathbb{R}^n$ and the vectors of length $\large n$. None of them are surprising or new. It seems silly to list them all out. But by carefully listing out what makes a vector a vector, we can see if other things work like vectors. If other things work like vectors, we can use all we know about linear algebra - distance, orthogonality, systems of equations, optimization, geometry, etc. - on those things!

We listed the properties we knew for $\large \mathbb{R}^n$ and vectors. We can now generalize to any set:

Definition: Let $\large V$ be a set on which addition and scalar multiplication are defined. If the 10 properties listed above (with $\large \mathbb{R}^n$ replaced with $\large V$) hold for all elements in $\large V$, then $\large V$ is called a vector space.

Two Quick Examples

Example of a Vector Space

Consider the set of all $\large 2\times 3$ matrices. Let's call it $\large M_{23}$. Is this set a vector space? Yes, all 10 properties above are satisfied. Here are a few of the properties:

1. The sum of two $\large 2\times 3$ matrices is definitely another $\large 2\times 3$ matrix, so $\large M_{23}$ is closed under addition.
2. Is $\large A+B=B+A$? Yup.
3. $\large cA$ is still a $\large 2\times 3$ matrix, so $\large M_{23}$ is closed under scalar multiplication.

We won't show all 10 properties (none of them are particularly interesting). But $\large M_{23}$ is a vector space, and that means we can treat $\large 2\times 3$ matrices as vectors.

Example of not a Vector Space

The set $\large \mathbb{Z}$ of all integers is not a vector space. Why not? The sum of two integers is an integer. Integers are commutative. To not be a vector space, it must fail at least one of the 10 properties. Which one(s) does it fail? It's #6. The set of integers is not closed under scalar multiplication. In other words, a scalar times an integer is not necessarily going to be an integer. For example, $\large \frac{2}{3}\cdot 4$ is not an integer.

Why Vector Spaces?

What have we done so far? We've listed the properties of vectors in $\large \mathbb{R}^n$ and given that list of properties the name vector space. Why would we do this? There are two big reasons this exercise ends up being useful.

Functions as Vectors

In a coming lecture, we will look at a few other vector spaces. The most useful application of the idea of a vector space is the fact that a set of functions can be a vector space. The idea of treating functions like vectors is mind blowing:

By treating functions as vectors, we can define things like orthogonality and distance for functions. This leads to big results in approximation of functions.

The theory of Fourier Series, which Mines students can learn about in classes such as MATH310, MATH455, and EENG411, is based on the ability to treat functions as vectors in a vector space!

Subspaces are useful

The primary application of vector spaces in this class is in the following definition:

Definition: A subspace of $\large \mathbb{R}^n$ is a subset $\large H$ with three properties:

1. $\large H$ contains the zero vector.
2. $\large H$ is closed under addition (the sum of any two vectors in $\large H$ is still in $\large H$).
3. $\large H$ is closed under scalar multiplication (if $\large \vec{v}\in H$, then $\large c\vec{v}\in H$).

Note

Many (most?) of the applications we'll see in the second half of this class are based on working with various subspaces of $\large \mathbb{R}^n$.

Example 1

Let $\large H$ be the $\large xz$ plane. Is this a subspace of $\large \mathbb{R}^3$? Let's see in the video below:

Example 2

Let $\large H$ be the set of all points on the line $\large x+3y=0$. Is this a subspace of $\large \mathbb{R}^2$? Let's see in the video:

Consider the example above. What if the line were $x+3y=2$ instead. Why do we no longer have a subspace?

Click to see the answer

 

In that case, the zero vector $\large \left<0,0\right>$ is not in the set, so it cannot be a subspace.

 

Note

Note that it is technically true that $\large \mathbb{R}^n$ is a subspace of $\large \mathbb{R}^n$ (go ahead and think about the definition). The set consisting only of the zero vector $\large \vec{0}$ is also a subspace of $\large \mathbb{R}^n$.

Section Outcomes

I can:

• Determine if a set is a vector space.
• Determine if a subset of a vector space is a subspace.