"The word proof is not something I'm fond of..." - gosh it means so much to hear a mathematician say that. Thank you for these videos, came back a second time because the title was funny 😂
0:23 : Definition 2:18 : Geometric Vectors 4:55 / 8:37 : IMPORTANT : The span of any number of vectors is a subspace ( a "mini" vector space on its own) 5:37 : proof 9:23 : Polynomials 11:09 : R3
At 11:36 you ask if the span of the two vectors is all of R^3. You then explain that it isn't because the second entry of both vectors is 0. Isn't it also true that a minimum of three vectors are needed in order to get a span that includes all of R^3? Isn't the largest possible span that can be achieved with two vectors a plane? Even if the second entry of one or both of the vectors was nonzero, wouldn't the largest possible span still be a plane? I'm thinking in terms of 3D geometric vectors and just want to confirm that the concept carries over to R^3.
All of your statements are correct, so it's a question of approach. Take a look at this video: ruclips.net/video/kz-RoBiqxRw/видео.html In the approach taken in these videos, ℝⁿ consists of sets of numbers and you can't apply terminology such as "plane". It's very important to learn to disassociate the two spaces so you can later re-associate them in a correct way. If you accept ℝ³ on its own terms, then it's not clear (at this point in the course) that two vectors is not enough and you need a more specific argument. If you learn to make the kind of argument described in this video, it will serve you well in many situations. That said, you *are* using the correct analogy. *Intuiting* things from the geometric space is indispensable. (But thinking of the two spaces as the same is going too far.)
Go to LEM.MA/LA for videos, exercises, and to ask us questions directly.
"The word proof is not something I'm fond of..." - gosh it means so much to hear a mathematician say that. Thank you for these videos, came back a second time because the title was funny 😂
0:23 : Definition
2:18 : Geometric Vectors
4:55 / 8:37 : IMPORTANT : The span of any number of vectors is a subspace ( a "mini" vector space on its own)
5:37 : proof
9:23 : Polynomials
11:09 : R3
The best linear algebra series I have watched so far. Thank you very much, Professor.
Mr Ginfeld, your videos are amazing.
thanks a million professor! The best linear algebra course
At 11:36 you ask if the span of the two vectors is all of R^3. You then explain that it isn't because the second entry of both vectors is 0. Isn't it also true that a minimum of three vectors are needed in order to get a span that includes all of R^3? Isn't the largest possible span that can be achieved with two vectors a plane? Even if the second entry of one or both of the vectors was nonzero, wouldn't the largest possible span still be a plane? I'm thinking in terms of 3D geometric vectors and just want to confirm that the concept carries over to R^3.
All of your statements are correct, so it's a question of approach. Take a look at this video: ruclips.net/video/kz-RoBiqxRw/видео.html
In the approach taken in these videos, ℝⁿ consists of sets of numbers and you can't apply terminology such as "plane". It's very important to learn to disassociate the two spaces so you can later re-associate them in a correct way.
If you accept ℝ³ on its own terms, then it's not clear (at this point in the course) that two vectors is not enough and you need a more specific argument. If you learn to make the kind of argument described in this video, it will serve you well in many situations.
That said, you *are* using the correct analogy. *Intuiting* things from the geometric space is indispensable. (But thinking of the two spaces as the same is going too far.)
Damn those pencils though. Packin' some serious heat
He probably uses them for vampires!
Nice explanation