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Current time:0:00Total duration:12:06

Video transcript

okay I have a vector V that's a member of RN so it's got n components in it so v1 v2 all the way down to VN I've touched on the idea but before but now that we've seen what a transpose is and we've taken transposes of matrices there's no reason why we can't take the transpose of a vector or a column vector in this case so what would be transpose look like well if you think of this as a n by 1 matrix which it is it has n rows and 1 column then what are we going to get we're going to have a 1 by n matrix when you take the transpose of it and this one column is going to turn into the 1 row so you're going to have it be equal to v1 v2 all the way to V N and you might remember we've already touched on this and a lot of matrices before and a lot of matrices let's say that some matrix a we called the row vectors of those matrix we call them the transpose of some column vectors a transpose a 1 transpose a 2 transpose all the way down to a n transpose is in fact not so many videos ago I had those row vectors and I could have just called them the transpose of column vectors just like that and that would have been in some ways a better way to do it because we've defined all these operations around column vectors so you could always refer to the transpose of the transpose and then do some operations on them but anyway I don't want to get too diverted but let's think a little bit of what happens when you operate this vector or you take some operation of this vector with some other vectors so let's say I have another vector here that's W and it's also a member of RN so if you have w1 w2 all the way down to WN there's a couple of things that we're already I think reasonably familiar with you can take the dot product of V and W V dot W is equal to what it is equal to v1 times w1 v1 w1 plus v2 w2 plus v2 w2 and you just keep going all the way to VN w n this is the definition of the dot product of two column vectors now how can we relate that to maybe the transpose of V well we could take we could take the transpose of V let me write it this way what is if I've you did a matrix multiplication so I did v1 v2 all the way to VN so this is V transpose that's V transpose and I take the product of that with W I take the product of that with W so I have w1 w2 all the way down to WN now if I view these as just matrices this is W right here I view these just as matrices is this matrix matrix product well-defined see over here I have a N by 1 matrix n by 1 matrix here I have a 1 sorry here the first one I have is a 1 by n matrix I have one row and n columns and here I have an N by 1 matrix I have n rows and only one column so this is well-defined I have the same number of columns here as I have rows here and I'm going this is going to result in a one by one matrix and what's it going to look like it's going to be equal to V 1 times W 1 V 1 times W 1 let me write it like this V 1 W 1 plus V 2 W 2 it's going to only going to have one entry we could write it as say just a 1 by 1 matrix like that let me just do it one by one matrix like that V 1 W 1 plus V 2 W 2 let me just I could write V 2 there plus all the way to VN WN that's it'll be it'll just be a 1 by 1 matrix that looks like that but you might notice that these two things are equivalent so we can make the statement we can make a statement that V dot W V dot W which is the same thing as WD these things are equivalent to V dot W is equivalent of let me just write it once over here V dot double you is the same thing as the transpose of V V transpose times W times W as just a matrix matrix product so if your view V is a matrix take its transpose and then just take that matrix and take the product of that with W it's the same thing as V dot W so that's an interesting take way I guess you could argue somewhat obvious and we've already been referring this do you know when I define matrix matrix products I kind of said you're taking the dot product of each row with each column and you can see that it really is it's really the dot product of the transpose of that row with each column but you got the general idea but let's see if we can build on this a little bit let's say I have some let's say I have some matrix a let me save our little outcome that I have there let's say I have some we get a good color here let's say I have some matrix a and it's an M by n matrix it's an M by n matrix now if I were to multiply that times a vector X so I'm going to multiply it by some vector X you can let's say that X is a member X is a member right this way X is a member of RN so it has n elements or another way you could view it is it's an N by 1 matrix now when I take the product of these what am I going to get or another way to say it is what is the vector ax when I take this product I'm just going to get another vector what's it going to be it's going to be an M by 1 vector this is going to be an M by 1 vector so we could say that ax is a member of our M it's going to have M elements right if this was equal to if you said that ax is equal to I don't know let's say it's equal to Z Z would have M elements you would have you would have Z 1 Z 2 all the way down to Z M and I know that because you have M rows in a and you have only one oh you could say this is this is M by n this is n by one you can the the resulting product will be M by one or it'll be a vector that is a member of our M it will have exactly M elements now if that's a vector of our M then the idea of dotting this with another member of our M is well-defined so let's say that I have another member of our M let's say I have a vector Y let's say Y is also a member of RM this has a the vector a X the vector that you get when you take this product as M elements this has m elements so the idea of taking there dot product is well-defined let me write that so we could take ax take that's a vector and now we are dotting with we are dotting it with this vector right here and we'll get a number we just take each of their terms multiply the corresponding terms add them all up and you get their dot product but what is what is this equal to what is this equal to we can just use this little I guess you could call a day a result that we got earlier on in this video using this result the dot product of two matrices or the sorry the dot product of two vectors is equal to is equal to the transpose of the first vector as kind of a matrix so you can view this as a X transpose this was a this is a M by 1 this is M by 1 this is M by 1 now this is now a 1 by M matrix and now we can multiply one by M matrix times y times y just like that now what is this thing equal to what is this thing equal to we saw a while ago I think it was two or three videos ago we saw that if we take the product of two matrices and take its transpose that's equal to the reverse product of the transposes you just switch the order and then take the transposes so this is going to be equal to this purple part is going to be equal to X transpose X transpose times 8 transpose times y times y this is just matrix products these are matrix products these aren't necessarily vector operations we're treating all of these vectors matrices and of course we're treating the matrix as a matrix so what is this equal to well we know that matrix products are associative you could put a parenthesis you know right now we have a parenthesis around there from there but we could we could just take another Association we could say that that is equal to that is equal to X transpose times these two matrices times each other this this is a vector but you can represent it as an M by one matrix so times a transpose a transpose Y a transpose Y just like that now let's think about what a transpose Y is let's think about it a transpose we have here a is M by n a is M by n what is a transpose a transpose is going to be n by M right it's going to be an N by M so this is an N by n M and then what is this vector Y going to be this is an M this is an M by one so when you take this product you're going to get an N by one matrix or you could imagine this as a vector that is a member of our n so this is a member of RN the entire product is going to result with a vector that's a member of RN and of course it's well-defined because this is a 1 1 by n vector right there now we can go back to our identity we have the transpose of some vector times some other vector they have the same well I guess you could say this has set as many horizontal entries as this guy has vertical entries just like that so what is this equal to we just use that identity this is equal to they're just regular x in this case instead of X transpose we'll just have X so this is equal to X X dot remember we just untranslated view it that way dot a transpose Y a transpose Y which is a pretty neat outcome we got this being equal to that we can kind of change the associativity although we have to essentially but change the order a bit and and take the transpose of our matrix so let me rewrite that just so that you can remember the outcomes are the two big outcomes of this video are I'll rewrite this one up here V dot W is equal to the matrix product of V transpose times W and if I have some matrix assume all of these matrix vector products are well-defined and all the dot products from L define if I have a X dot y dot Y some other vector Y this is equivalent to this is equivalent to X dot you get essentially putting the a with the other vector a transpose times y and this just might be a useful outcome or a useful result that we could build upon later in the linear algebra playlist