33 - The dimension of a vector space

what is the dimension of a vector space we gave a few clues in the previous lesson let's now define it formally so here's the definition of a basis we already know what a basis is it's a set of linearly independent vectors that's also a spanning set for the entire space ok so that's a basis and we've seen in examples that there could be many bases to the same space okay so here's first definition definition a vector space a vector space that has a finite basis a finite basis means that it has some basis with only finitely many elements is called a finite dimensional vector space okay so I'm defining what it means to be finite dimensional before I define the dimension okay so if you take a vector space for example r3 it has a finite basis we saw in an example we even saw two finite basis for our three then it's called a finite dimensional vector space okay and we still haven't defined wait wait wait I know what you're going to ask wait the answer is coming up wait in this course we are only gonna be looking at throughout the entire course at finite dimensional vector spaces all the theorems that we're going to state should actually start with the words let V be a finite dimensional vector space but since I'm saying it right now loud and clear I probably am going to forget to start all the theorems with the words let V be a finite dimensional vector space okay so everything in this course is going to relate to finite dimensional vector spaces except for a few examples that I that I'm going to give in a minute of vector spaces that are not finite dimensional just so you see that there are such things and in fact you know them they're not there I'm not going to invent anything that's new to you okay so a finite dimensional vector space if it has some finite basis good ok theorem and this theorem is a deep theorem it's not a trivial theorem in fact it relies on well the theorem itself is easy to prove if if you take for granted a deep lemma a deep theorem that it's based on and and so right now I'm just going to quote it we're going to discuss it a bit more later ok the theorem is let V well I am going to write it here be a finite dimensional vector space finding the finite dimensional vector space good everybody then any two bases a V have the same number of vectors okay so in the examples that we had for r3 we had two examples both had both were examples of Basie's for r3 both of which have had three elements right this is not a coincidence if one basis in r3 has three elements there could be many bases but all of them are going to have exactly three elements never - never four okay so this we're not proving now we are going to discuss it a bit later and now now that we know this theorem the definition the number of elements in a basis for V which is a finite dimensional vector space so if V is a finite dimensional vector space the number of elements in a basis does not depend on which basis you take it's an invariant that's what this theorem tells you okay the number of elements are the cardinality it's sometimes called so this number of elements is called the dimension of V and denoted denoted by dim good so what's the dimension of our three three okay clear so now we know what what the dimension of a vector space is for finite dimensional ones let's do a bunch of examples so here's example number one the dimension of our three equals three why because in the previous clip we saw a basis for R 3 we even saw two of them and they had precisely three elements good okay example two what's the dimension of F for any field for four right why you should be able to justify it's just that it's not just because the number four is sitting here why is it four well you have to go to the definition of the dimension you have to say okay it's four because I can show you a basis for F 4 F 4 is n tuples with four entries or not n tuples with four entries is ridiculous it's four tuples with entries in F right what is a basis for this exactly the standard basis 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 and 0 0 0 1 that's a basis for F 4 it has four elements therefore that I mention is for good clear ok what is the dimension what is the dimension of the vector space are n of X these are polynomials of degree less than or equal to n over the field are what's the dimension n plus 1 n plus 1 why why it's not just a general knowledge it's it's a very solid fact we saw a basis for this vector space the basis was the set of polynomials 1 X x squared X cubed all the way to X to the N ok we argued that it's indeed a basis that they're linearly independent and that they span everything it wasn't that tricky Wright was pretty much obvious but it has n plus 1 elements good ok for now please don't jump with the answer think and raise your hand if you think you know it so what's the dimension of matrices with M rows and n columns over um let's say R what's the dimension of the vector space of matrices with M rows and n columns M by n matrices what's the dimension yes very good M times n why why if you want to like it purple we could translate it into an end temple and then double the size n times n we did this in the previous lesson okay so we need to show you need to demonstrate a basis a basis for this space that has M times n elements that's what you need to do right so for example for example let's maybe do first a toy example where m is small and n is small okay so eg let's look at um let's look at M 2 by 3 over R what is a basis a basis is going to be the following matrices 1 0 0 0 0 0 do you agree 0 1 0 0 0 0 and allow me to add dot dot dot H dot corresponding to a matrix and finally 0 0 0 0 0 1 do you see 6 matrices here and do you see that any 2 by 3 matrix can be written as a linear combination of these a general 2x3 matrix is ABCDE F it's 8 times this plus B times this plus C times this and so on right does everybody see that okay and they're linearly independent how do you know they're linearly independent for example straighten them out as rows in a big 6x6 matrix and you're going to get the identity matrix right away no 0 rows good so this is the in fact the standard basis for this space okay and it can be generalized and it has 6 elements 2 times 3 right do you see that ok and this can be generalized um maybe let's let's write it this goes here and this goes here so let's write it in general in fact in fact I'm even going to change this R to a general field F to a general field F and the reason I want to write it more generally is because there's a bit of notation of standard notation here and I want to mention it so for so this is an example this is an example in general in general we write let's do this just so I can write it more so let's take V to be this M by n matrices over the field F and then the statement is that then V equals M n okay that's what we want to show so let's take a basis for V so a basis for V is usually denoted by E I J j e IJ where so the set of these e I J's is a basis for V where I have to tell you what do I mean by E I J where e IJ is a matrix that is all zeros all zeros except a single one sitting in the I row and J DH column that's a IJ and all the rest is zeros clear do you see that it's precisely this so 4 M 2 by 3 over r this is e 1 1 this is e 1 2 this is e 2 3 see that ok so this is just notation do you agree ok do you agree that this is a basis just like here ok I'm not going to write all the formal details but it's exactly the same sort of arguments good ok um okay here's another exact questions about this is this is this clear general okay go ahead earlier when we gave examples for bases mm-hmm we use the example for our 3 1 1 0 0 1 0 1 right 0 1 1 so could I take 1 0 0 at 0 1 1 and find that as a basis so yeah ok so the question is we had a previous example in which we found a basis for R 3 that had three elements namely 1 1 0 0 1 0 1 and 0 1 1 now you're asking maybe two of them would form a basis without the third one right I want to modify them a bit I want to say that 1 0 0 and 0 1 1 would also be a basis but I if you are you taking just two elements I'm taking two different elements but you're asking if only two can form a basis well first of all the answer has to be no because of the theorem that I quoted that any basis for our 3 is going to have 3 elements that's first of all now second of all you're asking ok why is my specific example so you're asking why isn't this 1 0 0 and this sorry 0 1 1 a basis for our 3 the answer is it doesn't span r3 you can only get vectors of the four abb you would never get a vector of the form ABC where the B and the C are different so this set it there indeed linearly independent but they don't span all of our three good okay any more questions on these four examples everybody good okay so here's example number five R of X R of X is not is polynomials with one variable X over the field R but with no limit no limitation on the on the degree of the polynomial okay so all the polynomials form a vector space right you can check all the axioms they're all satisfied right but what can you say about the dimension of this right it's not finite dimensional is not a finite dimensional vector space now it seems obvious right you're saying hey clearly I can't find a a finite number of polynomials that span every single polynomial right but the fact that you say clearly I can't find well maybe some alien can show up from a different planet it said hi you missed it this is a 30 1529 dimensional space here are the polynomials that you need to take that would span everything okay so in order to prove that it's not a finite dimensional vector space you have to somehow argue that there cannot be a finite basis for this space okay and why can't there be even if some very advanced life-form are fives from a different planet why is he necessarily wrong and there cannot be a finite basis for all the polynomials well if there's a finite basis then it has finitely many polynomials in it one has the maximal degree one of them has the maximal degree of a finite set one has the maximal degree do you agree and then you can never generate a polynomial with a higher degree than that by linear combinations of those polynomials right so if the highest degree is 1 million and there are other polynomials take linear combinations of them you would never get a polynomial of degree greater than 1 million right because the that the mult that a scalar multiple of a polynomial does not cannot raise its degree and adding polynomials cannot raise the degree two you agree okay so I'm not going to write all that down just tell me if you understood if it's clear yeah that would be you have to write everything I just said without the stuff about the alien and and that would be the the actual proof that R of X cannot have a finite basis okay so if it were to have a finite basis then there would be a maximal degree and you cannot generate anything with a higher degree contradiction it's not a basis good okay here's another example another example so we had our n of X I just want to add F n of X and maybe spell out in fact we did this okay let's not write it it's it's I just wanted to maybe generalize look here at this example so do you agree that if I replace R by F it would be the same thing where the the basis is just the set 1x x squared all the way to X and plus good okay so questions about the dimension of a vector space is it clear what is the dimension of a vector space why it's well-defined okay even though it's the number of elements in any basis and there could be more than one basis that number is fixed okay and that depends on the on a theorem that we mentioned but didn't prove okay so what I want to do next is discuss some theorems regarding uh dimensions and bases okay and and of course they're gonna relate to linear independence and spanning and all those properties because that's how these notions were defined okay

Channel: Technion

Views: 48,238

Rating: 4.8447762 out of 5

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

Id: d-W82f52mTM

Length: 21min 57sec (1317 seconds)

Published: Thu Nov 26 2015