Vector Spaces | Definition & Examples
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- Опубликовано: 1 апр 2025
- Vectors in R^n obey a list of rules, things like commutivity of vector addition that a+b = b+a as vectors. However, we can abstract this list of rules and introduce the general concept of a vector space. A vector space has elements that might be all kinds of weird things, but as long as they obey the same list of rules, they are a vector space. And the key here is that all the things we have done in linear algebra extend to this new situation too! Our first new example is polynomial of degree less than or equal 2, which have a natural "vector" addition and scalar multiplication.
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That's a great video, learned a lot, you're doing a great job, but could anyone please help me?
At 02:32 the second equation in the green group, shouldn't that be:
(c+d)u = cu +du
@@DrTrefor hello sir, it's been a real honor for me that you replied me, love from India!
No its fine
You explained in 8 minutes what my college teacher couldn't explain in 1 semester.
Excellent work!
Me too
You are like my dream teacher. The straightfoward explaination is extremely relaxing. I'm not scared that a hodgepodge of terms will be thrown to my face. Thank you very very much for making this.
Wow, thank you!
Yeah and so handsome
Duuuuuude! I'm writing a test in a few minutes and you just saved my life once again.
This channel is truly underrated.
This playlist is amazing , I will watch all of your videos!!!
One of my favorite teacher ❤
After watching the English tutorial, you explained all the key points of vector space linear transformations abbreviated easily.
I'm from Bangladesh 🇧🇩
Your class was so much helpful.
Thank you 🖤
i like your energy when u teach
Thank you for your hard work, Trefor. This video series is very helpful and much appreciated!
I appreciate your hard work, always getting to understand more from your explanations
You've mentioned at least twice in the video that translation between different vector spaces (specially from Rn to other vector spaces) would be of great use and importance; So I wish you provided an example of such translation as well. But this is very helpful, makes some things feel less abstract to me. Thanks.
That's fair. A bit later in the playlist is videos on the polynomial vector space which is our first really new one, and this has a direct translation
@@DrTrefor Great, I had no idea this was part of a playlist, I've found it now. Really cool playlist!
Instantly subscribed! Keep up the good work!
I wonder if you know how many students lives you're saving. THANK YOU
Thanks for the clear explanation. Love your energy. And you have beautiful eyes.
Amazingly clear explanation!
Best explanation ever!
THANK YOU SO MUCH!
i came here, to get the feel of vector space, i already know all these rules, everyone is doing it, i just can't visualize what i am writing. That's the problem
Excellent teaching
Excellent content. U just need to improve the audio quality and organise content more maybe in a web site or re arrange videos in playlists in cleaner way
If you could generate transcripts for these videos, it will be really helpful !!
Dr . Trefor … thankuh so so so much ❤
Hi Trefor! Just to reaffirm my understanding, when you defined vector spaces, it is not only defined for vectors (like what we’ve seen so far) but anything that obeys the rules at the beginning of the video (in this case, the example of polynomials)?
(if anyone still needs the answer to this) I'm pretty sure that vector is used to refer to any element of a given a vector space, for example, a polynomial of degree n could be considered a vector of a vector space of polynomials of degree n (the degree of the vector space could be greater though, a polynomial of degree 3 could be located in a vector space of polynomials of degree 5, similarly to how an arrow vector with a z coordinate of 0 is a vector of 3d space despite seeming like it fits in 2d space, it depends on the context you view it in). Since a vector is just a member of a vector space, that would mean that your assumption is correct as a vector space is any set that follows those rules
You voice is very deep, so you'd better use a microphone which can cancel echo. It will be wonderful.
And talk a bit slower.
For me it's fine
Hi Professor, if my understanding is right, vectors in R^n (for example, lemme take R^2, in 2 dimensional space) which are a set of vectors which obey these rules like commutative, vector addition and scalar multiplication form a space called vector space??
Very helpful.
are there any videos in this series about the kernel and range of a linear transformation?
Can you do a. Video on how 2x2 matrix is a vector space or 4x8 matrix
Hello sir please recommend a book of linear algebra
I feel you are saying a good things, but can you turn on or add(I don't know) the translation 😚
Love from core of my heart
3:36 that's the face I make when I try to understand math
haha:D
if any 2 arbitrary vectors in R^2 space obey these 10 vector addition and multiplication rules, is that mean these 2 arbitrary vectors forms a vector space or the span of these 2 vectors as vector space ?
Nice
What’s the point with vector space? How is it different?
Need help s={1,x1,x2,.....,xn} Is this vector space of R³?🙏
Doesn't contain the zero vector, so no
@@DrTrefor but x1,x2,...,xn are element of real number
Would you kindly add generated subtitles sir? I am not native speaker
Matrices are a vector space ?
bravooo
Why mathematicians don't explain why division is not part of an operation of the vector space?
You can divide by any scalar by multiplying, if you wanna divide a vector by 3 then just multiply it by 1/3.
On the other hand "multiplying" a vector by a matrix, while called multiplication, is actually a linear combination.
Similarly, "multiplication" of matrices, while called multiplication, is actually a composition of matrices/transformations
There are as many divisions as there are multiplications, it's just that they named a lot of different stuff "multiplication" ...
Can someone explain vector without i, j, k? It sickens me.
Agreed brother. It would make it way more relatable
PIEASieans Sir Shahid ko kaho iski videos chala liya karen