Subspaces are the Natural Subsets of Linear Algebra | Definition + First Examples
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- เผยแพร่เมื่อ 5 ต.ค. 2024
- A subspace is a subset that respects the two basic operations of linear algebra: vector addition and scalar multiplication. We say they are "closed under vector addition" and "closed under scalar multiplication". On a subspace, you can do linear algebra! Indeed, a subspace is an example of a vector space. We see that all of R^n, {0}, and lines through the origin are all subspaces, but that lines NOT through the origin are not subspaces.
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This is by far the most elucidated linear algebra course series that I've ever taken.
All my previous teachers were not actually teaching linear algebra, they were just teaching computational rules and never explained the notion or the idea behind it even in a graduate school level, so, it was hard for me to relate linear algebra with other realms of science.
Thank you!
Glad they are helping!
Thank you! My linear algebra teacher just introduced this topic using theorems and didn’t even explain what it represented. This helped!
Glad it was helpful!
Dear education system,
Please make sure your seatbelt is on, the best teachers in the world are now able to teach the entire planet.
It's ridiculous how good Dr Trefor is ; )
This is such a captivating introduction into why we use subspaces (because we now have vectors in our life) and why they must follow the 2 closed rules. Thank you for this great video!
i have search a lot on youtube and thanks god , finally i got you sir .
This is an awesome explanation! Great visualization.
Glad you liked it!
Thanks god I found your channel, you deserve more subs
This Is a Great TEACHER!!!
Thank you sir
It makes the concept more clear and easy
This is absolutely wonderful explanation of vector spaces !!
Man… it clarified so many things for me.
such a helpful video that is explained in such an organized, enthusiastic and pedagogical manner! Thanks for the help!!
Your channel is so underrated
Awesome explanation. Thanks Dr.Trefor !
Thank you so much for this wonderful explaination.
Thank you so much for this amazing explanation about subspaces! It is really helpful!
I love your work! Thanks a lot for all your help here.
Happy to hear that!
This man is a truly hero.
Thanks for the concise explanation.
Really great explanation, love from India
Thanks your doing the world a service.
Thank you for this awesome video!
thanks a lot sir. also thanks miss ism who shared this video :)
Great video
thank you
you are awesome!
Very helpful! Thank you.
Thanks
Thanks so much!!
In the subspace example of a line not passing through the origin, we took the two vectors x and y not belonging to S subspace ,but shouldn't we take x and y vectors belonging to S and then check the conditions.
Thank you Sir❤❤❤❤
You're most welcome!
May i know, what is the application of subspace? 🙏🙏 I hope someone can answer me cause i desperately need the answer
thank you bro
Are the vectors v = (−1,3,2) and w = (2,−2,4) orthogonal in R4?
It has been two years
I'm thinking should I answer or not
😂
So will a subspace always have at least 1 fewer dimensions than the original space? If the vector space is R2 then must the subspaces be 1 or 0 dimensions?
If the spaces is a space of functions, say polynomials of degree
It is a theorem that all the sub space of two dimensional plane are the origin, lines through the origin, and the entire plane itself.
@@DrTrefor sure enough, ran into that in my text (Applied Linear Algebra by Olver and Shakiban) not long after watching this vid.
Hi Professor Trefor, I’m a little confused. In the first line example with the line that passes through the origin and satisfies all conditions (around 3:30ish), you said that if we add vector x to a vector y it will be on the green line so I guess still an element within the sub space. But aren’t we assuming that x and y in this case are just scalar multiples of each other? What if they weren’t? Then no combination of x and y would fall on the line? Alternatively, if S is defined to be the entire Cartesian plane, then I guess we could then say that it satisfies all three conditions? I’m sorry if my question is long and confusing!
if subspace just on that green line ->my opnion
My teacher just taught numericals so I didn't understood 😂
Thank u
so it means that there are only two possible subspaces that are possible in 2d and one is the zero and the other is the whole xy plane
Every subspace of R5 that contains a nonzero vector must contain a line. Is this statement true?
I wonder if subset contains the vector 0 or the subspace of 0 then the subset itself a subspace ? Thank you!
Thank you🙌🏾💙
Thanks for the tutorial. Going to your example of the line (subspace) that does not go through the origin, you satisfy the 1st property by multiplying any vector on the line by zeros (is that right? has to be at least two zeros, depending on n dimensions?) you get the zero vector. I'm having trouble seeing this. This scalar multiplication certainly gives you a zero vector which is (0, 0), but how is this in the subspace? The origin does not lie on this line, so it actually still fails the first test. The way I see this, is that you are actually creating a zero vector on the line, but how do you describe this with coordinates? Do you define it as a vector (or point) on the line with no initial point, only a terminal point? Or vice versa?
@@DrTrefor I misunderstood the second property completely. Multiplying any vector on the line by the scalar zero gave you a zero vector, and I thought this 'somehow' satisfies the first property. I realized that even though we get a zero vector, it is outside this line, so the first property remains unsatisfied. This is an epiphany of sorts after thinking about this for two days. Thanks.
Thanks
I still don’t get it :(
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