34 - Properties of bases

okay so I want to give you a few theorems of properties of faces that are going to be they're very basic theorems very elementary and we're going to use them a lot so here goes so let V be finite dimensional I remember to write it by the way some of these theorems are going to be true even for non finite dimensional vector spaces but I don't care right now we're interested only in finite dimensional so yeah so I'm supposed to right now TF AE oh well I'll capitalize it t f AE this stands for the following are equivalent the following are equivalent meaning that I'm gonna list now three things each of which implies the others so it's a it's a big if and only if if and only if statement okay so one B is a basis for V 2 B is a maximal maximal linearly independent set so by a maximal linearly independent set what I mean that it's a linearly independent set da but that its maximal in the sense that it cannot be contained in a bigger linearly independent set you throw in even one more guy that's it it's not linearly independent okay so that's what what I mean by being a maximal linearly independent set and three B is a minimal spanning set by which I mean that it's a spanning set it spans all of V if you throw away even one guy it doesn't span anymore okay so it's as small as can be as a spanning set okay okay so TF AE the following are equivalent 1 implies 2 2 implies 1 2 implies 3 3 implies 2 1 implies 3 3 implies 1 each 2 are equivalent ok in order to prove this how many things do I need to prove good but I wanted somebody to first say 6 and then somebody to say 4 and then you to say that ok so a priori I need to prove 6 things right that each two are if and only if but in fact if I prove 1 implies 2 and 2 implies 3 then for free I get that 1 implies 3 do you see that if I also prove that 3 implies 1 then I get that any two are equivalent right if I prove 1 implies 2 implies 3 implies 1 3 things I get the other 3 for free because you go from any other two via one of the others do you see that ok so in order to prove this proof I need to prove three things and here goes the first thing I'm going to prove is that 1 implies 2 so why does 1 imply 2 1 said that B is a basis right by definition it means okay so we know 1 B is a basis by definition it's linearly independent and it spans everything by definition B is linearly linearly independent right we want to show that it's maximal linearly independent namely that it cannot be constrict Li contained in a linear in the then set okay so B is linear independent we need to show that it's maximal if we add an element to B okay so we want to we want to see if if we can make it bigger while still maintaining that property of being linearly independent okay so if we add an element to B then if we add an element let's call it W to B then W is a linear combination of the elements of B why I took B I know it's linearly independent I want to show that it's maximal linearly independent so I threw in one more guy any guy I'm claiming no matter who you threw in any W that you added that's it W is a linear combination of the others why no no no no it's written here right because B is a basis so aside from being linearly independent it's a spanning set right so any W you add is going to be a linear combination of the others do you agree okay so by definition B is linearly independent and if we add an element W to B then W is a linear combination of the elements of B since B is a spanning set do you agree therefore be together with W is no longer linearly independent therefore B was maximal linearly independent you cannot add anything without ruining the linear independence therefore B is maximal linearly independent in that sentence good so if you know it's a basis its maximal linearly independent okay let's prove that two implies three I don't know why I have these parentheses around them makes it mysterious two implies three so now my assumption is that we're looking at a maximal linearly independent set and what I need to prove that it's a minimal spanning set okay so let's do that if B is maximal linearly independent then any element any element any vector any W that we add to B if its maximal linearly independent than any W that we add would make it dependent if its maximal linearly independent that means that when we add something it's no longer linearly independent that we add to B we'll make we'll make it dependent do you agree okay if it's maximal independent throw in one more guy that's it it's no longer independent okay no longer independent means it's dependent that means that there's somebody that's a linear combination of the others okay that somebody cannot be any of the original elements in B because V was independent that somebody has to be W do you agree good let's write that therefore there is right intuitively W is extra but we can write it more precisely more accurately therefore there is an element which is a linear combination of its predecessors oh boy you compile this word for me again I think I got it right by now and that has to be W if it's linearly dependent suppose the original elements were v1 through VA and now we added W if it's linearly dependent one of the guys is a linear combination of its predecessors it can't be the V's because they were linearly independent it has to be W so no matter who you tried to add no matter who you try to add you get that it's a linear combination of the others and evac any vector is a linear combination of those V is do you agree so that implies hence B spans you took somebody W any W threw it in and saw the dump of the use a linear combination of the elements of B so it's bins do you agree but that's it it was a linearly independent set it spans right so now we know that it's a spanning set are we done no we have to so show that it's a minimal spanning set okay so hence B spans V we need to show we need to show B is minimal minimal in the sense that if you take somebody away you no longer span everything okay so if we remove a VI from B so B let's say was v1 through VN au you take one of the V eyes away what does it mean for it to be minimal it means that it no longer spans everything right if we removed VI I claim that it no longer spans everything tell me somebody that's not in the span exactly if VI were in the span it would mean that VI is a linear combination of the others but it can't because it was linearly independent set good so if we remove the eye from B be no longer spans V since VI is not a linear combination of the other V J's cuz it should were then the original set B would be dependent one is a linear combination of the others clear so this proves that 2 implies 3 if its maximal linearly independent implies its minimal spanning okay finally we need to show that 3 implies 1 and that would complete the proof ok so now we know that we're starting with the minimal spanning set a minimal spanning set we have to show that it's a basis ok so if B is minimal spanning well in particular it's spinning right so what remains to be shown is that it's linearly independent right so if it's linear if it's minimal spanning B is spanning right it's a spanning set remains to show that it's linearly independent so how do we show that it's linearly independent if it's not if it's not linearly independent so if B is dependent if B is dependent it means that there's somebody extra there's somebody there that's a linear combination of the others right but then we can remove it and still have a spanning set which contradicts the minimality do you agree so let's write that if B is dependent then there is a VI which is a linear combination of V J's right there's one guy there that can be written as a linear combination of the others but then but then B without VI would still be spanning but still be a spanning set spanning set in contradiction to minimality contra the two minimality good so the proof seems like it's kind of long and but in fact it's just the terminology if you remember the definitions what it means to be a basis what it means to be a spanning set what it means to be linearly independent what it means to be linearly dependent then everything becomes really obvious really obvious and if you don't remember the definitions you can stare at this for three hours and it just won't compile okay so if you want to understand this proof and you're feeling a bit fuzzy go read the definitions that we had for all these terms that we're using again and again and again being a spanning set being where is it linearly independent gone be a linear combination being linearly independent and linearly dependent where are these words on the previous board okay so read the definitions and then read this and you'll see that everything is really straightforward though this is really the only place where we applied a previous theorem which was also very basic okay but really we just use the definitions okay so there are more properties of BC's that I want to discuss but our time for today is up so Wilkin - in in our first lesson next time questions everybody good okay

Channel: Technion

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Keywords: Technion, Algebra 1M, Dr. Aviv Censor, International school of engineering

Id: Sa68cBN8fbY

Length: 17min 44sec (1064 seconds)

Published: Thu Nov 26 2015