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Current time:0:00Total duration:9:32

Null space 3: Relation to linear independence

Video transcript

so I have the matrix a over here and a has m rows and n columns so we could call this an M by n matrix and what I want to do in this video is relate the linear independence or linear dependence of the column vectors of a to the null space of a so first of all what am I talking about is column vectors well as you can see there's n columns here and we can view each of those as an M defect M a dimensional vector and so let me do it this way so you could view this one right over here we could write that as V 1 V 1 this next one over here this would be V 2 V 2 and you would have n of these because we have n columns and so this one right over here would be V n V V sub n and so we could rewrite a we could rewrite the matrix a the M by n matrix a and bolding it to show that that's the matrix we could rewrite it as so let me do it the same way so draw my little brackets there we can write it just express it in terms of its column vectors is we could just say well this is going to be V 1 for that column V 1 for that column V 2 for this column all the way we're going to have n columns so you're going to have V n for the nth column and remember each of these are going to have M terms or I should say M components in them these are M dimensional column vectors now what I want to do I said I want to relate the the linear independence of these vectors to the null space of a so let's remind ourselves what the null space of a even is so the null space of a the null space of a is equal to or I could say it's equal to the set it's the set of all vectors X that are members of our N and I'm going to double down on why I'm saying RN in a second such that such that if I take my matrix a if I take my matrix a and multiply it by one of those X's by one of those X's I am going to get I am going to get the zero vector so why do this why why does X have to be a member of RN well just for the matrix multiplication to work for this to be if this is M by n let me write this down if this is M by n well in order to just make the matrix multiplication work or you could say the matrix vector multiplication this has to be an N by 1 and n by 1 vector and so it's going to have n components so it's going to be a member of RN if this was em by a well or let me use a different letter if this is M by I don't know seven then this would be R 7 that we would be dealing with so that is the null space so another way of thinking about it is well if I take my matrix a and I multiply it by some vector X that's member of this null space I'm going to get the zero vector so if I take my matrix a which I've expressed here in terms of its column vectors multiply it by some vector X so some vector X and actually let me make it clear that it doesn't have to have the same so some vector X right over here draw the other bracket so this is the vector X and so it's going to have it's a member of RN so it's going to have n components you're going to have X 1 as the first component X 2 and go all the way to X and if you multiply so if we say that this X is a member of the null space of a then this whole thing is going to be equal to the zero vector is going to be equal to the zero vector and once again the zero vector there's going to be an M by one vector so you're going to it's going to look she let me write it like this it's going to have the same number of rows as a so I'll try to make it the brackets roughly the same length so and there we go try and draw my brackets neatly so I'm going to have M of these one two and then go all the way to the M zero so let's actually just multiply this out using what we know of matrix multiplication and by the definition of matrix multiplication one way to view this if you were to multiply our matrix a times our vector X here you are going to get the first column vector V 1 V 1 times the first component here X 1 X 1 plus the second component times the second column vector X 2 times V 2 V 2 and we're going to do that n times so plus dot dot dot X sub n times V sub n V sub N and these all when you add them together are going to be equal to the zero vector now this should be this so it's going to be equal to the zero vector and now this should start ringing a bell to you when we looked at when we looked at linear independence we saw something like this in fact we saw that these vectors V V sub 1 V sub 2 these n vectors are linearly independent if and only if any linear if and only if the solution to this or I guess you could say the weights these vectors the only way to get this to be true is if X 1 X 2 xn are all equal 0 so let me write this down so V sub 1 V sub 2 all the way to V sub n are linearly independent linearly independent if and only if if and only if only solution so let me only solution or you could say weights on these vectors to this equation only solution is x1 x2 all the way to xn are equal to 0 so if the only solution here if the only way to get this this sum to be equal to the 0 vector is if X 1 X 2 and X all the way through xn are equal to 0 well that means that our vectors v1 v2 all the way through VN are linearly independent or vice-versa if they are linearly independent then the only solution to this if we're solving for the weights on those vectors is if for x1 x2 and xn to be equal to 0 remember linear linear independence if you want to say that it's still mathematical but a little bit more common language is if these vectors are linearly independent that means that none of these vectors can be can be constructed by by linear combinations of the other vectors or looking at this way this right over here is a you could view this as a linear combination of all of the vectors that the only way to get this linear combination of all the vectors to be equal to 0 is if x1 x2 all the way through xn are equal to 0 and we proved that in other videos on linear independence well if the only solution to this is all of the X ones through xn sir is equal to 0 that means that the null space this is only going to be true you could say if and only if the null space of a the null space of a let me make sure it looks like a matrix I'm going to bolt it the null space of a contains one vector it only contains the zero vector remember this is if all of these are going to be zero well then the only solution here is going to be the zero vector is going to be is going to be the zero vector so the result that we're showing here is if the column vectors of a matrix are linearly independent then the null space of that matrix is only going to consist of the zero vector or you could go the other way if the null space of a matrix only contains the zero vector well that means that the columns of that matrix are linearly independent