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Current time:0:00Total duration:10:41

More on matrix addition and scalar multiplication

Video transcript

in the last video we started off with two linear transformations we had the linear transformation s that was a mapping from RN to RM and then we had the linear transformation T that was also a mapping from RN to R M and we defined the idea of the addition of two of these two transformations so s plus T this transformation of X we defined as being equal to s of X this vector plus plus T plus T of X and of course this will be in R this input is still from RN and then this each of these our vector is in RM if we add two vectors to R in RM to each other we get another vector in RM because RM is a valid subspace it's also closed under addition so this is still a mapping so s plus T s plus T is still a mapping from RN to R M and we also said that look every linear transformation we've shown in a previous video can be represented as a matrix that we could say that s of X is equal to some matrix a times X and we could also say that T of X is equal to some matrix B times X and these both of these would be M by n matrices they and let me write that M by and both of these because these are both mappings from RN to RM and what we did is we made another definition we defined this was a definition right here then we made another definition we defined the addition of two matrices we said the matrix a this is any matrix a plus B they both have to have the same dimensions same dimensions so they're both M by n in this case and we defined this addition to be a new matrix where each column of this matrix is the sum of the corresponding columns of these matrices so this matrix matrices this matrix is first column will be the sum of a first column in B's first column so a1 plus b1 the second column I'll do a little line here is a2 plus b2 and it goes all the way to a n plus BN this was a definition and the whole reason why we made this definition is because if you defined matrix addition in this way then this thing this thing when you replace it with ax plus BX you get to that this thing over here is equal to its the corresponding matrices by this definition a plus B times X that was this was the motivation this was the motivation to get to a nice expression like this for defining matrix addition in this way now that's this all seems very abstract so let's actually add a matrix or let's add two matrices so we'll start off with a 2 by 2 case so let's say I'm adding the matrix 1 3 minus 2 4 2 the matrix form they have to have the same dimensions to the matrix 2 7 minus 3 minus 1 what do we get well by definition you just add up their corresponding columns you add up the first column when you add up of course funding columns what happens when you add up two columns to two vectors well you just add up their corresponding entries so essentially when you add up two matrices you're just adding up all of the corresponding entries I'll talk about it in this way just because that's how I defined it but it's they're all equivalent so let's look at the first thing the first column and this matrix right here is going to be this column Plus this column so it's going to be 1 plus 2 let me write it like this 1 plus 2 and then minus 2 minus 3 minus 2 minus 3 and then the second column that one right there is going to be 3 plus 7 3 plus 7 and then 4-1 4-1 and so this will be equal to two three ten minus five and three just like that and notice even though the definition is I'm adding up corresponding columns but what in effect happened well I'm just adding up the corresponding entries I added the one to the two the three to the seven the minus 2 to the minus 3 the 4 to the minus one it's that straightforward nothing fancier than that in fact we could have we could have rewritten this definition we could rewritten this definition if we if we say that the vector or the matrix a is equal to you know a 1 1 a 1 2 all the way to a 1 N and then if you go down this is a 2 1 all the way to a + 1 and then you go all the way down there to a n n we've seen that before and then the matrix B just by the same argument or but you know similar definition is this is B 1 1 that entries B 1 1 that's B 1 2 all the way to B 1 n this is B 2 1 second row all the way down to B n sorry this is M we have M rows so this is M n so this right here is B M 1 this would be B M 2 all the way down to this is B M n right there I want to be very careful these are M by n matrix matrices so this this row down here is the M row in both cases but we could redefine our matrix or another way of stating this matrix addition definition is you say look if I'm going to add a plus B I'm going to add a plus B I'm just going to add up the corresponding entries so this entry up here is going to be in a different color it's going to be a 1 1 plus B 1 1 this one's going to be a 2 1 plus B 2 1 all the way down to a M 1 plus B M 1 and then this is going to be of course a 1/2 plus B 1 2 all the way to a 1n let me scroll over a little bit all the way over to a 1 n plus b1n and then all the way down to a m n plus b MN these are equivalent definitions this just this takes a lot less space to write it I felt comfortable doing it because we've already defined mate vector addition but it essentially boils down to you just add up all of the corresponding entries that's all matrix addition is it's it's probably one of the simplest things that you've seen in in your recent mathematical experience now matrix scalar multiplication very similar idea we defined we defined scalar multiplication times of transformation of X to be equal to a scalar times the transformation of X this was a definition this was a definition and we also defined we also defined scalar times some matrix a to be equal to the scalar a new matrix where each of its columns are the scalar times the column vectors of a so a1 and then the next column is C a2 and then you go all the way to C see a N and the whole motivation for this was because this could be simplified to well T I've said was equal to BX the transformation sorry this is eight times transformation of X the transformation T of X was equal to so this we still have our C so it's going to be C times the matrix B times the vector X that's what the transformation of X could be written as and so this would be equal to and so if I use my this would be equal to by just manipulating and we did this in the last video by actually breaking this up in the column vector is multiplying them by each of the components of X and then distributing the C and rearranging a little bit we can now say using this definition that this is equal to some new matrix CB we're using this definition some new matrix CB where it's essentially C times each of the column vectors of B times X this right here was our motivation we wanted to be express this as a product of a some new matrix and a vector because any linear transformation should be expressible in that way and that's why we made this definition now let's apply it and I wanted to show you that this is this is perhaps even simpler than matrix addition so if we want to multiply the scalar five times the matrix I'll do a three by two matrix so one minus one two three seven zero this will just be equal to by this definition I'm just saying look I'm multiplying the scalar times each of the column vector so it's five times one to seven but what's that that's just five times each of the entries so it's five times 1 which is 5 5 times 2 which is 10 5 times 7 which is 35 then the next column is going to be 5 times this column right here which is just 5 times each of its entries so 5 times minus 1 is minus 5 5 times 3 is 15 5 times 0 is 0 it's that simple you literally if we go back to this definition we can define scalar multiplication of a matrix so we could also define CA as being equal to if we say this is a representation for a of the scalar C times each of the entries of a that's it so it's C times a 1 1 C times a 1 2 all the way to C times a 1 n you go down this way C times a 2 1 all the way down to C times a em 1 and then if you go down the diagonals B C times a MN you literally just take your scalar and multiply it times every entry in a and that's all you have to do so hopefully this clarified things up a little bit or maybe it was a bit of review from things you might have learned in high school