This video is (to me) the summation of all the material that has come before, and is the single most important material I have so far seen on linear algebra anywhere. If you patiently work through all of the material before this, and absorb this video, you walk away with a really sound understanding of the column space and the null space, and their critical relationship. I really wish I had seen this a long time ago!
Around 7:11 you erase the trivial null space and replace it with the first non trivial entry. However, shouldn't the null space always include the trivial zero vector, or is it assumed and by convention not written at all when you add non-trivial entries representing linear dependence?
+The Flagged Dragon Great question! Actually, after the change, the zero vector is still captured by the expression: when alpha=0, so it's still in the mix.
I came here from your worksheet, "Matrix from column space and null space", hoping for a hint. Your instruction is incredible. (edit) PS: I got the answer, thanks!
The relationship dim R + dim N = # cols "technically" breaks down at 4:25 IF we view the zero vector [0] as a 1X1 matrix, with one column and one row ! In this limiting case, dim R + dim N = # cols leads to 1 + 1 = 1 :) However, this is rectified if we do NOT view the zero vector in N as a 1X1 matrix with dimension 1 (because 1 column in N is counted) but rather as a dimensionless vector of zero dimension.
The zero vector by itself is definitely a vector space of dimension 0. The question is: is the set consisting of the zero vector alone linear independent? The answer is "no" because there's a nontrivial linear combination that equals the zero vector. That combination is 1*(zero vector) = zero vector. So everything works cleanly and, in fact, this is a very important special case to always check for every general linear algebra statement.
17:15 ...So the column space is the subspace spanned by the set of linearly independent column vectors? And there is a solution (called b) to the linear system if b is in the span of the column space? I can think of two other ways to say this: 1. A linear combination of the vectors in the column space can be used to generate b OR, 2. that b is linearly dependent on the vectors in the column space. Trying to use all this terminology properly, can someone clarify and help me make this more precise if I've made a mistake?
Could we also say that the column space is capturing linearly independent relationships and the null space is capturing linearly dependent relationships?
How do we know that we didn't miss any fancy zeros in the null space? Why is it that "adding" one linearly dependent column to the matrix corresponds to adding only one dimension to the Null space (and not more)?
What a brilliant explanation. I finally get the concept rather than just mechanically solving the problems. By the way can you give an intuitive explanation as to why is the null space important?
Null space is particularly important to capture all the possible solutions a linear system have or a decomposition problem have in case of problem have infinitely many solutions.
How does the row space play into the mix? If the null space lives in r9 (in this example) so does the row space. I'm curious to know your thoughts on their relationship! Great video!
The classes and the teacher’s teaching are excellent, fantastic! But I don't understand why some videos are without subtitles, not even in English. For non-native English speakers, understanding is very difficult. Could you fix this, @MathTheBeautiful?
Go to LEM.MA/LA for videos, exercises, and to ask us questions directly.
Wow. Feels like watching the mathematics evolve organically. Utterly beautiful lecture!!!
This video is (to me) the summation of all the material that has come before, and is the single most important material I have so far seen on linear algebra anywhere. If you patiently work through all of the material before this, and absorb this video, you walk away with a really sound understanding of the column space and the null space, and their critical relationship. I really wish I had seen this a long time ago!
100% agreed
Really well done Pawel! Great exposition of the two spaces.
Bravo! never seen it explained so clearly. the ability to make complex things seem trivial and obvious is the mark of a true master.
What an exceptional beauty this lecture series happens to be. V carefully planned, methodical and systematic!!!
Thank you very much for your videos. I simply love the way you approach the subject!
very insightful, thank you !
This is a fantastic lecture, and beautifully brought together these two concepts together. Thank you so much! :) :) :)
Thank you for your commitment to teaching the subject
Around 7:11 you erase the trivial null space and replace it with the first non trivial entry. However, shouldn't the null space always include the trivial zero vector, or is it assumed and by convention not written at all when you add non-trivial entries representing linear dependence?
+The Flagged Dragon Great question! Actually, after the change, the zero vector is still captured by the expression: when alpha=0, so it's still in the mix.
+MathTheBeautiful Oh!! I see! Good point. Thank you for the response
I came here from your worksheet, "Matrix from column space and null space", hoping for a hint. Your instruction is incredible.
(edit) PS: I got the answer, thanks!
Well done! I'm sure you enjoyed it!
The relationship dim R + dim N = # cols "technically" breaks down at 4:25 IF we view the zero vector [0] as a 1X1 matrix, with one column and one row !
In this limiting case, dim R + dim N = # cols leads to 1 + 1 = 1 :)
However, this is rectified if we do NOT view the zero vector in N as a 1X1 matrix with dimension 1 (because 1 column in N is counted) but rather as a dimensionless vector of zero dimension.
The zero vector by itself is definitely a vector space of dimension 0. The question is: is the set consisting of the zero vector alone linear independent? The answer is "no" because there's a nontrivial linear combination that equals the zero vector. That combination is 1*(zero vector) = zero vector. So everything works cleanly and, in fact, this is a very important special case to always check for every general linear algebra statement.
@@MathTheBeautiful Kindly allow me to thank you for your complete answer. By the way, your videos are real gems!
17:15 ...So the column space is the subspace spanned by the set of linearly independent column vectors? And there is a solution (called b) to the linear system if b is in the span of the column space? I can think of two other ways to say this: 1. A linear combination of the vectors in the column space can be used to generate b OR, 2. that b is linearly dependent on the vectors in the column space. Trying to use all this terminology properly, can someone clarify and help me make this more precise if I've made a mistake?
Could we also say that the column space is capturing linearly independent relationships and the null space is capturing linearly dependent relationships?
That's a good way of putting it. Although is it really a relationship if it's independent?
How do we know that we didn't miss any fancy zeros in the null space? Why is it that "adding" one linearly dependent column to the matrix corresponds to adding only one dimension to the Null space (and not more)?
What a brilliant explanation. I finally get the concept rather than just mechanically solving the problems. By the way can you give an intuitive explanation as to why is the null space important?
Null space is particularly important to capture all the possible solutions a linear system have or a decomposition problem have in case of problem have infinitely many solutions.
How does the row space play into the mix? If the null space lives in r9 (in this example) so does the row space. I'm curious to know your thoughts on their relationship! Great video!
Thank you sir
Sir,could you proof or explain that null space is linear independence ?
Although through the observation it tells the answer.
super
The classes and the teacher’s teaching are excellent, fantastic! But I don't understand why some videos are without subtitles, not even in English. For non-native English speakers, understanding is very difficult. Could you fix this, @MathTheBeautiful?
Please accept my apologies with regard to the lack of subtitles. I just don't have the bandwidth to do this.
I understand the column space but I did not get the null space