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Current time:0:00Total duration:10:40

Video transcript

we spent a good deal of time on the idea of a null space what I'm going to do in this video is introduce you to a new type of space that can be defined around a matrix it's called a column space column space and you could probably guess what it means just based on what it's called but let's say I have some matrix a now let's say it's an M by n matrix so I can write my matrix a and we've seen this multiple times I can write it as a collection of column vectors so this first one second one and I'll have n of them how do I know that they I have n of them because I have n columns and each of these column vectors we're going to have how many components so v1 v2 all the way to VN well they had this matrix has m rows right so each of these guys are going to have M components so they're all members of our M so the column space is defined as all of the possible linear combinations of these column vectors so the column space of a this is my matrix a the column space of that is all the linear combinations of these column vectors well what's all of the linear combinations of a set of vectors it's the span it's the span of those vectors so it's the span of vector 1 vector 2 all the way to vector N and we've done it before when we first talked about span and subspaces but it's pretty easy to show that the span of any set of vectors is a legitimate subspace it definitely contains the zero vector if you multiply all of these guys by zero which is a valid linear combination added up you'll see that it contains the zero vector if let's say that let's say I have some vector a that is a member of that some member of let's say it's a member of the column space of a that means it can be represented as some linear combination so a is equal to you know c1 times vector 1 plus c2 times vector 2 all the way to C n times vector n now the question is is this closed under multiplication is if I multiply a times some new let me call it let me see I multiply it times some scalar s I'm just picking a random letter so s times a is this in my span well s times a would be equal to s c1 v1 + s c2 v2 all the way to s CN VN which is once again just a linear combination of these column vectors so this si would clearly be a member of the column space of a and then finally to make sure it's a valid subspace and this actually doesn't apply just to column spaces this applies to any span this is actually a review of what we've done in the past we just have to make sure it's closed under addition so let's say a is a member of our column space let's say B is also a member of our column space or our span of all of these column vectors then B could be rewritten could be written as I don't know let me say v1 times v1 plus b2 times v2 all the way to BN times VN and my question is is a plus B is a plus B a member of our of our span of our column space the span of these vectors well sure what's a plus B a plus B a plus B is equal to c1 plus b1 times v1 times v1 plus c2 plus b2 times v2 right I'm just literally adding this term to that term to get that term this term to this term to get this term and then it goes all the way to VN plus CN times VN which is clearly just another linear combination of these guys so this guy is definitely within the span so the span of and and this what I just did it doesn't have to be unique to a matrix I mean a matrix is just really just a way of writing a set of column vector so this applies to any span so this is clearly a valid subspace so the column space of a is clearly a valid subspace now let's think about other ways we can interpret this notion of a column space let's think about it in terms of let's think of it in terms of the expression let me get a good color if I were to multiply my let's think about this let's think about the set of all the values of if I take my M by n matrix a and I multiply it by any met any vector X where where X where X is a member of remember X has to be a member of RN it has to have n components in order for this multiplication to be well-defined so X has to be a member of RN let's think about what this means this is the set if this says look I can take any member any N component vector and multiply it by a and I care about all of the possible products that this could equal all the possible values of a X when I can pick and choose any possible X from RN well let's think about what that means ax ax if I write a like that and if I write X like this let me write it a little bit better let me write X like this x1 x2 all the way to xn what is ax well ax could be re-written as x1 and we've seen this before ax is equal to x1 times v1 plus x2 times v2 all the way to plus all the way to plus xn times VN right we've seen this multiple times this comes out of our definition of matrix vector products now if ax is equal to this if ax is equal to this and I can I'm essentially saying that I can pick any vector X and RN I'm saying that I can pick all possible values of the entries here all possible real values in all possible Asians of them so what is this equal to what is the set of all possible so I could rewrite this statement here as the set of all possible X 1 V 1 plus X 2 V 2 all the way to xn VN where X 1 X 2 all the way to xn are member of the real numbers that's all I'm saying here this statement is the equivalent of this when I say that X the vector X can be any member of RN I'm saying that its components can be any members of the real numbers so if I just take the set of all of the essentially the combinations of these column vectors were their real numbers where we're there where their coefficients are members of the real numbers what am i doing this is all the possible linear combinations of the column vectors of a so this is equal to the span sorry well it equals the span of v1 v2 all the way to VN which is the same thing which is the same exact same thing as the column space of a so the column space of a you could say hey what what are all the possible vectors or the set of all vectors I can create by taking linear combinations of these guys or the span of these guys or you can view it as what are all of the possible values that ax can take on if X is a member of RN so let's think about it this way let me let's say that I were to tell you that a that I need to solve the equation ax ax is equal to let me say well the convention is to write a B there but let me put a special B there let me put B 1 let's say that I need to solve this equation ax is equal to B 1 and then I were to tell you let's say that I were to figure out the column space of a and I say B 1 not is not a member of the column space of a now what does that tell me that tells me that this this right here can never take on the value B 1 right because all of the values that this can take on is the column space of a so b1 is not in this it means that this cannot take on the value of b1 so this would imply this would imply that this equation we're trying to set up a X is equal to b1 has no solution has no solution if it had a solution if it had some solution so let's say that ax equals b2 so let's say ax equals b2 has at least one solution has at least one solution what does this mean well that means that this for a particular X or maybe for many different X's you can definitely achieve this value right for there's some X's when you multiply it by a you definitely are able to get this value so this implies that B 2 is definitely a member of the column space of a some of this stuff on some level it's almost obvious this comes out of the definition of the column space the column space is all of the linear combinations of the column vectors which another interpretation is all of the values that a X can take on so if I try to set a X to some value that it can't take on clearly I'm not going to have some solution if I set ax if I am able to find if I am able to find a solution I'm if I am able to find some x value where ax is equal to B 2 then B 2 definitely is one of the values that a X can take on anyway I think I'll leave you there now that you have at least a a kind of abstract understanding of what a column space is in the next couple of videos I'm going to try to bring everything together what we know about column spaces and null spaces and whatever else to kind of understand a matrix and a matrix vector product from every possible direction