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The set of all subspaces of a vector space V forms a partially ordered set (poset) under set inclusion (⊆). The smallest subspace is {0}, and the largest subspace is V itself. Lattice Structure: A lattice in mathematics is a poset in which any two elements have a well-defined: Greatest lower bound (GLB) or meet, denoted A∩BA∩B for subspaces. Least upper bound (LUB) or join, which corresponds to the span of the union of two subspaces, denoted span(A∪B). Subspaces of a vector space do indeed form a lattice under these operations: Meet (∩) is the intersection of two subspaces. Join (∨) is the span of their union. Terminology: While subspaces form a lattice, they are rarely called "a lattice" in standard linear algebra courses. The term "lattice" is more commonly associated with lattice theory or abstract algebra. In contexts where lattice theory intersects with linear algebra, this lattice of subspaces might be explicitly studied as a modular lattice (because it satisfies the modular law).

should we use a different set of symbols or braces for a multiset, so we can distinguish it from a set

at 45:57 i would call it 0_n (or even better 0_nxn) for the zero matrix, not Z_n. The latter symbol Z_n is usually reserved for the ring Z_n

Watching 30:44 , i think keeping the last pair of parentheses might be better because it emphasizes the nested scope of the existential/universal quantifiers. e.g ∃x∈ℝ [ ∃y∈ ℝ ( xy=1 ) ] . Also I think it is a little cleaner - for readabilty sake - to parenthesize the quantifiers themselves. e.g. (∃x∈ℝ) [ (∃y∈ ℝ) ( x * y = 1 ) ] .

Thanks. I love it too!

I love this way of teaching, using the original textbook and elucidating or expanding on it.

I noticed that you definition of polynomial is different than some algebra texts that I am reading where they define it by a function from the natural numbers into the ring in which they have an Evalution map. Does it really matter, how I define it?

In what lecture was Theorem 1.3 proved?

The book is excellent, it's to the point and truly is quite comphresive

Professor, at 42:13 talking about fields, in particular the definition given, a) and b) should be exactly like that of in a ring no? because in that definition we don't necessarily state that a) and b) hold for the zero element which it should. So shouldnt that definitioln use for a and b just F and not F*

I leave it here: I will spend the rest of my life filling out the details.

Hello. I wonder if you're familiar with Harold Edwards's book on Galois theory, where he sticks close to what Galois himself actually presented rather than the modern, more abstract approach that came later. For someone like myself who's interested in the historical development of the subject, I'm curious if you feel Edwards's book would be a good starting point.

49:00 Will any alpha do?

38:40. The composition beta o alpha in the initial category C has domain B and codomain D, while that composition in the Comma category has domain f and condomain h. Can two morphisms be the same even if they have different domains and codomains?

26:48 Why do you make those dual statements logically equivalent? You just said dual statements are not generally so. Shouldn't it be: Pi implies p in C iff Pi^op implies p^op in C^op?

1:02:41 To my mind f , as defined there, is an isomorphism in both the categorical and the noncategorical sense; in the categorical sense, it is an isomorphismo because g exists s.t. gof = 1_P and fog = 1_Q. No mention of structure preservation is made in the categorical definition.

"Important and relevant" sums up every aspect of these lecture courses! Other courses have made LA feel like a very narrow and overgrown path, kinda easy to see the way forward but impossible to go anywhere else, not without getting bombarded with terminology that was never properly explained in context. I think what you have done is very valuable. I am very grateful that you decided to make such learning material freely available.

Oh he stopped making videos ...

12:34 Wait??? This can't be right. He probably meant a proper class can't be an element of a class. Is that it? Because a class can definitely be an element of another class, considering a set is a class and sets can be elements of sets ...

👍

Thank you so much professor

nice. I'm watching the whole series !

Yo whaaaaat? The man, the myth, the legend. Gonna get at these lectures ASAP

the big picture of searching for canonical forms including the Shurs and Spectral theorems, was very inspiring!! thank you.

Sir can you upload these colored notes as i want to print them .they are very good notes

hello sir .... when will you complete the 5th volume .. order and lattices ?

I barely understand the proof for 9.25. Is it that we simply assume that G/Hp is Lagrangian the entire time?

Thank you very much . Your series are perfect

Thank you very much . The series are perfect

professor aεp is correct in upper bound and lower bound definitions

Pax ave et vale

I almost finish this series. You’re my hero, professor. Thank you, professor！

Danke. Is this book available?

I need youre pdf please 😢

Hi sir . if i partition 3 people into 2 indistinguishable groups of size 2 and 1, then the answer would be C(3,2)C(1,1)/2!=1.5. Where did i do wrongly? Your series of videos is the best i’ve seen so far,Thank you,sir.

The best professor

Even though I'm on vacation following your wonderful clips.

Hi. Professor, what is your correct email address? This one dedekindt-sr.. does not work

It would be a great idea to make a lattice theory series. Great content 👌

Hey, I'm a PMATH student from the University of Waterloo, Canada. Do you advise taking commutative algbebra before Algebraic Geometry?

you’re welcome

Thanks professor!

Thanks professor!

Thanks professor!

I would like to thank you so much for this clear course, Professor S. Roman. I would also like to inquire whether the cancellation theorem holds true for semi-direct products.

Very good class

Still watching

At 7:51 the prime factorization of that number should have a 6 as the exponent of 5 instead of what was there. I'm assuming this mistake has been corrected since but just a note to any viewer who was confused.

Dear Professor Roman, thank you for your detailed explanation on this topic. Sorry if my question is silly. You mentioned that the presence of zero vector destroys minimality, but not the essentially unique representation of a set of vector. Does it contradict to theorem 1.54 which shows that a minimal spanning set also provides essentially unique representation for its span. In other words, let me assume that I have a set of vectors (including 0) that provides essentially unique representation (since addition of 0 doesn't change this property), is this set a minimal spanning set or not?

This is too blurr to see