Vector Spaces and Subspaces (A Step-by-Step Guide)

What if I told you that vector spaces and subspaces are like the continental United States and the great state of Colorado?

Jenn (B.S., M.Ed.) of Calcworkshop® teaching vector spaces subspaces

Jenn, Founder Calcworkshop^®, 15+ Years Experience (Licensed & Certified Teacher)

Analogy: Vector Spaces as the United States

Think of it like this.

The United States is huge!

With bustling cities and towns, dense forests, sprawling deserts, rolling hills, spacious skies, amber waves of grain, and purple mountain majesty … just borrowing a few words from the song America, the Beautiful.

Each region and state are unique, thus, making it difficult to show or “prove” its splendor.

So instead of trying to prove every aspect of what makes the United States exceptional, which would be quite an extensive list, we will choose just one state to focus on.

Because if we can show that this one state is amazing by association, we have thus proven that all 50 states are also amazing.

That’s the idea of vector spaces and subspaces. The continental United States is the vector space containing 48 connecting states (vectors).

Colorado is the subspace, a smaller vector space within the larger space. If we can show that Colorado has all the properties we need and want, then that means the continental United States does as well.

Vector Space Subspace (US Map)

Axioms of Vector Spaces

Okay, what qualities or properties do we need to prove?

You see, a vector space is a nonempty set $V$ of vectors (objects) on which two operations, addition, and scalar multiplication, are defined and subject to ten rules or axioms.

The axioms hold for all vectors $\vec{u}, \vec{v}$ , and $\vec{w}$ in $V$ and for all scalars $c$ and $d$ :

The sum of $\vec{u}$ and $\vec{v}$ , denoted $\vec{u} + \vec{v}$ , is in $V$ .
$\vec{u} + \vec{v} = \vec{v} + \vec{u}$
$(\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w})$
There is a zero vector $\vec{0}$ in $V$ such that $\vec{u} + \vec{0} = \vec{u}$
For each $\vec{u}$ in $V$ , there is a vector $- \vec{u}$ in $V$ such that $\vec{u} + (- \vec{u}) = \vec{0}$
The scalar multiple of $\vec{u}$ by c, denoted $\vec{c u}$ , is in $V$ .
$c (\vec{u} + \vec{v}) = c \vec{u} + c \vec{v}$
$(c + d) \vec{u} = c \vec{u} + d \vec{u}$
$c (d \vec{u}) = (c d) \vec{u}$
$1 \vec{u} = \vec{u}$

Proving Whether a Set is a Vector Space

And to prove whether $V$ is a vector space, we must demonstrate that all ten axioms hold, or we must find a counterexample showing that $V$ is not a vector space.

An Example of a Set Not Being a Vector Space

For example, assume we let $V = {[\begin{array}{l} x \\ y \end{array}] : x \geq 0, y \geq 0}$ . Is $V$ a vector space?

To show that $V$ is a vector space, we must show that addition and scalar multiplication is preserved via the ten axioms. But, if we put our thinking caps on, we quickly realize something is amiss.

Knowing that $V = [\begin{array}{l} x \\ y \end{array}] = ⟨ x, y ⟩$ is any vector such that $x \geq 0, y \geq 0$ , this limits us to the first quadrant where $x$ and $y$ are both positive.

Graphing Vector Space (First Quadrant)

The addition of two vectors, where $x$ and $y$ are positive, will also result in a vector where both elements are positive. Therefore, $V$ is closed under addition.

Vector Space (Closed Under Addition)

However, scalar multiplication is another thing entirely for $V$ . Remember, $c$ is any real number, which means we can multiply our vector by a negative number.

What would that mean?

Suppose we let $c = - 2$ .

Then, $- 2 [\begin{array}{l} x \\ y \end{array}] = ⟨ - 2 x, - 2 y ⟩$ which when graphed on the xy-plane puts the vector in quadrant III, as shown below.

Scalar Multiplication (Not Vector Space)

But we’re only allowed to be in quadrant $I$ , as $x \geq 0, y \geq 0$ . Consequently, $V$ is not closed under scalar multiplication, and we have just found a counterexample showing $V$ is not a vector space.

Introduction to Subspaces

But gosh, what would happen if $V$ is a vector space?

We would have to prove all ten axioms!

And no one wants to do that!

So, instead of proving all ten, we will prove a subspace with only three axioms.

Again, think… if we can prove Colorado (subspace) is great, and if Colorado is inside the continental United States, then this proves that the United States (vector space) is also great.

A subspace of a vectors space $V$ is a subset $H$ of $V$ that has three properties:

The zero vector of $V$ is in $H$ , so it’s not empty.
$H$ is closed under vector addition. That is, for each $\vec{u}$ and $\vec{v}$ in $H$ , the sum $\vec{u} + \vec{v}$ is also in $H$ .
$H$ is closed under multiplication by scalars. That is, for each $\vec{u}$ in $H$ and each scalar $c$ , the vector $c \vec{u}$ is in $H$ .

Now, we can all agree that proving three axioms is better than 10, but how do we begin?

The Idea of Span and Subspaces

Here’s the trick: a subspace is just like the span.

The $Span {{\vec{v}}_{1}, \vec{v_{2}}, \dots, \vec{v_{p}}}$ is the set of all vectors that can be written as linear combinations $\vec{v_{1}}, \vec{v_{2}}, \dots, \vec{v_{p}}$ , such that:

$Span {\vec{v_{1}}, \vec{v_{2}}, \dots, \vec{v_{p}}} = c_{1} {\vec{v}}_{1} + c_{2} {\vec{v}}_{2} + \dots + c_{p} \vec{v_{p}}$

Think of it like a subset.

If a subset of $R^{n}$ is any collection of points in $R^{n}$ , then a subspace is a subset of the vector space.

Therefore, given any subspace $H$ of $V$ , a spanning set (i.e., generating set) for H is a set ${{\vec{v}}_{1}, \vec{v_{2}}, \dots, \vec{v_{p}}}$ in $H$ such that:

$H = Span {{\vec{v}}_{1}, \vec{v_{2}}, \dots, \vec{v_{p}}}$

Proving a Set is a Vector Space

Let’s look at an example.

Suppose $W$ is the set of all vectors of the form $(3 a + b, c - 2 a, 5 b - c)$ .

Where $a, b$ and c represent arbitrary real numbers, and we want to show that $W$ is a vector space.

$\begin{aligned} W & = (3 a + b, c - 2 a, 5 b - c) \\ = [\begin{array}{c} 3 a + b \\ c - 2 a \\ 5 b - c \end{array}] \end{aligned}$

First, let’s write the vectors in $W$ as column vectors.

$\begin{aligned} W & = [\begin{array}{c} 3 a + b \\ c - 2 a \\ 5 b - c \end{array}] \\ = [\begin{array}{c} 3 a + b \\ - 2 a + c \\ 5 b - c \end{array}] \\ = a [\begin{array}{c} 3 \\ - 2 \\ 0 \end{array}] + b \underset{v_{1}}{\underset{⏟}{[\begin{array}{c} 1 \\ 0 \\ 5 \end{array}]}} + c \underset{v_{3}}{\underset{⏟}{c [\begin{array}{c} 0 \\ 1 \\ - 1 \end{array}]}} \end{aligned}$

Now, let’s check to see if the zero vector can be found by letting $a, b$ , and $c$ be zero.

$\begin{aligned} a [\begin{array}{c} 3 \\ - 2 \\ 0 \end{array}] + b [\begin{array}{l} 1 \\ 0 \\ 5 \end{array}] + c [\begin{array}{c} 0 \\ 1 \\ - 1 \end{array}] & \to 0 [\begin{array}{c} 3 \\ - 2 \\ 0 \end{array}] + 0 [\begin{matrix} 1 \\ 0 \\ 5 \end{matrix}] + 0 [\begin{array}{c} 0 \\ 1 \\ - 1 \end{array}] \\ = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}] \end{aligned}$

Since zero vector is in the vector space, we can now see if the set of vectors spans W by looking for a consistent matrix using elementary row operations.

$\begin{aligned} [\begin{array}{ccc} 3 & 1 & 0 \\ - 2 & 0 & 1 \\ 0 & 5 & - 1 \end{array}] & \sim [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] \end{aligned}$
Therefore, $W$ is a vector space as set ${[\begin{matrix} 3 \\ - 2 \\ 0 \end{matrix}], [\begin{array}{l} 1 \\ 0 \\ 5 \end{array}], [\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}]}$ spans W.

See, it’s easy!

Next Steps

In this lesson, you will:

Learn how to prove vector spaces
Find counterexamples
Find a set of vectors that span a vector space
Determine how many vectors are in the span
Check if W is in the subspace spanned by the vectors or by the columns of matrix A

Get ready to explore these concepts, and let’s get started!

Video Tutorial w/ Full Lesson & Detailed Examples

vector-spaces-and-subspaces-example

Get access to all the courses and over 450 HD videos with your subscription

Monthly and Yearly Plans Available

Get My Subscription Now

Still wondering if CalcWorkshop is right for you?
Take a Tour and find out how a membership can take the struggle out of learning math.

4.9 / 5

1,128 Reviews

Kevin S.10-30-24 - CO, United States

I had never even heard of discrete math before I had to take it for my Bachelor's in Cybersecurity. I had actually hoped that my math days were done! I suffered through class not understanding any of the presentations until I stumbled across Calcworkshop. I used the Calcworkshop lessons more than I used my instructor's lectures and my book. I ended up with a high B (just 12 points away from an A) thanks to Jenn. Without her easy to understand style of teaching, I'm not sure I would have even passed the class. Thank you so much!!!

Richard10-29-24 - Norway

Very well structured and site. It is well structured giving a good overview of the different topics regarding all mathematics where you easily can choose which path is applicable to your specific needs.The video lectures are superb with introduction to the topic followed by a detailed theoretically explaination and what I personally get most out of is; heaps of examples where you are taken and shown, step by step, the prosess of solving the problem. They also includd practice sheets and solutions to test yourself after each chapter. There is a lot to undertake on this site, but it provides an easy and understandable interface to map out what you need to do. The site has been very helpfull for me and I highly recommend it.

RobWP10-26-24 - United States

I am taking this course for mainly linear algebra review. It is a course that I have had but fairly long ago. I'm currently working in masters in data science and linear algebra namely transformations is something that has come up and I thought I need a good review on. I so far what I have liked about the course is, the common sense approach. Her approach makes it easy to remember the concepts because of her common sense approach. I like the fact that she gives information that is not really in any textbooks. I can vouch for that because not only have I used textbook recently to review I have had a formal class in the past that used textbooks. She goes over things thoroughly and for the price it's very much worth it. I have had many quarters of formal calculus and differential equations courses and I am kind of anxious to look to see what her approach to calculus is. I can't really answer how this class would be for a novice or someone who has not had linear algebra the past, but I really don't think they need to worry about that. Her method I feel will make a novice be successful and to understand the material. She is very easy to understand. Gives good examples but doesn't beat a dead horse either. What I like about her introductory remarks is that she says what linear algebra is all about and she points out what the core of the linear algebra is about and what you need to know. Having taken it before I would have to agree and she is dead on spot.

Robbin03-18-25 - CO, United States

Very well organized, helpful and comprehensive. It is great also to have access to prior course material so you can easily go back and refresh if you need to. Jenn is very clear and the number of examples leaves one comfortable with each topic. Couldn't be happier.

RCJK03-07-25 - ID, United States

I had to take Calculus I and II as accelerated classes while working full time. It was my first time taking this courses and was very overwhelming. Calcworkshop really made learning the subjects easier for me by having all the subjects easy to find and easy to understand explanations. I received a B in both classes which I am extremely happy with doing the classes in double time.

Jacqueline S.03-07-25 - SC, United States

Jenn is awesome!Her breakdown of the topic is beyond what you would expect. She is showing you how to arrive at the answer instead of doing a simple equation. Most people need to know the why it give confidence and structure to the learning process. This is what Jenn is great at, she knows how to deliver the information to a non math mind in a way that instills knowledge that you can forever draw from as you move forward. It's really well thought out. She is a true master teacher. Thanks for all that you do for us! Hugs n Blessings... JS

Rose M.03-07-25 - CA, United States

Best instructor ever, I have never had Math explained in a way that was so interesting. Her lectures and instruction is explained perfectly, and I will continue to use this for all my future math courses.

A Reviewer03-07-25 - United States

I needed help with college precalculus, they provided more than enough help that allowed me to succeed and get the scores I needed.

Jujuana03-05-25 - MO, United States

I love the videos the way each lesson is explained step by step. I love that she has examples that you can go over again and again in order to understand the way algebra works. I was lost because my instructor was flying through each lesson and I was totally lost. I attached my syllabus and they helped me stay focus on what I was working and give me a guide to go back and refresh what I went over in class.

Stacey01-31-25 - AZ, United States

Hello, I highly recommend getting a subscription to Jenn and her Calcworkshop teaching math service. I did a general, "go from bad to good at math", Google search and found out about Calcworkshop through responses to the topic on reddit. I've just started watching her videos and found them to be extremely helpful. I'm going back to college after a long absence and math was never my strong suit. So I'm starting at the very bottom of her available subjects and going from there. Just watching a couple of the videos has taken me from being dragged along not understanding to following along just fine in my math class. I'm moving from the "I'll never get this", to the "I look forward to understanding this", mindset in approaching math. This for me is huge, and even though I have a long way to go, I'm confident I will get there thanks to Jenn. Totally worth it!

Kathy W.01-14-25 - WA, United States

Calcworkshop is an outstanding source to learn math and understanding math concepts. Jenn is an outstanding master teacher. Every lesson is clearly presented making understanding easy to grasp. Practice lessons with accompanying answers are given for each section. I am 82, a retired teacher and lifelong learner. Learning calculus has always been on my bucket list. Calcworkshop is helping me reach my goal.

Cheryl01-14-25 - NY, United States

Calc workshop is the absolutely best resource for math and I have seen almost everything that exists out there. I have even recommended it to my daughter's school so that all the kids could use the videos during the summer months for the subject they plan to study in the fall. They are guaranteed to get a A if they have watched the videos and completed the problems before taking the class. Students who are weak in areas can access lower level classes and those who seek to advance above grade level can do so. The videos and exercises are crystal clear and the excellent presentations raise the bar for teachers everywhere. I have always believed that one great math teacher is important in the education process. The videos, exercises and problems are excellent. It is not possible to have ANY knowledge gaps. I have never seen anything on the market that even begins to compete with it!

Justin N.12-22-24 - OK, United States

Calculus can be a scary subject. I know I was scared. I had been out of any math class for over 23 years. I had taken an 8 week Calculus 1 class and felt like I was overloaded. Going into Calculus 2 I found CALCWORKSHOP.com. I used the training videos and material to really dive back into Calc 1 and prepare for Calc 2. Long story short with the material provided and ample study time, I finished Calc 2 with a 95%. Thanks Jenn!

Shelby12-10-24 - OH, United States

CalcWorkshop was a wonderful resource as someone going onto a college calculus 1 class after not having taken a math course in a year! Jenn always explained things in a logical order that built upon previous lessons, and the way she explained topics was always easy to understand. Calcworkshop saved this semester for me, and I couldn't be more grateful to have found this site. :)

Jordan Q.12-09-24 - AR, United States

Anyone struggling with math at any level would greatly benefit from this product. I signed up for it immediately after getting a D on my first exam in calculus 2 and ended up getting A's and B's on the remaining exams, getting a B overall in the class. Jenn breaks down the topics, without skipping any steps, in a way that makes them very easy to understand. I highly recommend this product!

N8Booker11-28-24 - MA, United States

I'm in school to get my degree Networking and Cybersecurity and I'm in my first year. I've been away from school for quite a long time so the concepts that are being taught although not foreign to me, have so much dust on them from a lack of being used in real world applications regularly.I stumbled upon CalcWorkshop by pure chance and it has been a game changer for me. Jen filled in the gaps in between what my Professor is teaching at a rapid fire pace and where my brain grasps and applies it. If not for Jen's approach, I'd have failed my test I took miserably!Am I grateful? It goes beyond words and not I recommend her workshops to everyone!

Kevin S.10-30-24 - CO, United States

Richard10-29-24 - Norway

RobWP10-26-24 - United States

RCJK03-07-25 - ID, United States

Jacqueline S.03-07-25 - SC, United States

See more reviews on Shopper Approved