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Current time:0:00Total duration:19:50

Visualizations of left nullspace and rowspace

Video transcript

in the last video I had this 2x3 matrix a right here and we figured out all of the subspaces that are associated with this matrix we figured out its null space its column space we figured out the null space and column space of its transpose which you can also call the left null space and the row space or what's essentially the space spanned by a's rows but let's write it all in one place because i realize it got a little disjointed and see if we can visualize what all of these look like especially relative to each other so let me copy and paste my original matrix no copy and then let me scroll down here and paste it over here and it pastes let me see if I can find our key takeaways from from the last video so our column space right here of a was this thing right here let me write this this was our column space it was the span of the r2 vector 2 4 we copy that copy that bring it down paste this was our column space let me write that this is the column space of a was equal to that right there and now what other things do we know we know what the well we know that the left null space was a span of two one let me write that so our left null space or the null space of our transpose either way it was equal to the span of the r2 vector 2 1 2 1 just like that and then what was our null space our null space we figured it out in the last video here it is it's the span of these two our three vectors we copy and paste that so that's edit copy let me go down here let me paste it so that was our null space right there and then finally what was our row space what was our row space or the column space of our transpose the column space of our transpose so the column space of our transpose was the span of this r3 vector right there so this one right here and so let me copy and paste it copy and scroll down and we can paste it just like that and just like that okay let's see if we can visualize this now now that we have them all in one place so first of all if we imagine a transformation if we imagine a transformation X that is equal to a times X our transformation is going to be a mapping from what X would be a member of r3 so r3 would be our domain so it would be a mapping from r3 and then it would be a mapping to r2 because we have two rows here right you multiply a 2x3 vector a 2 by 3 matrix times a 3 by 1 times a 3 by 1 vector and you're going to get a 2 by 1 vector so it's going to be a mapping to r2 so that's our codomain so let's draw our domains and our Co domains so you have I'll just write them very generally right here so you can imagine r3 is our domain domain and then our codomain is going to be our 2 it's going to be our two just like that Co domain and our T is a mapping or you could even imagine a is a mapping between any vector there and any vector there when you multiply them now what is our column space of a our column space of a is the span of the vector 2 minus 4 it's an R 2 vector this is a subspace this is a subspace of R 2 we could write this let me write this so our column space of a these are just all of the vectors that are spanned by this we figured out that these guys are just multiples of this first guy or we could have done it the other way week I said this guy's mult this guy and that guy are multiples of that guy either way but the is just one of these vectors we just have to have one of these vectors and so it was equal to this right here so the column space is a subset of r2 and what else is a subset of r2 well our left null space our left null space is also a subset of r2 our left null space is also a subset of r2 so let's graph them actually so I won't be too exact but you can imagine let's see if you if if we have if we draw the vector 2 4 let me draw some axes here let me scroll down a little bit so if you have some vector let me draw am I do this as neatly as possible so my vertical axis that is my horizontal axis and then what does the span what does the span of our column space look like what does the span of our column space look like so you draw the vector 2 minus 4 so you're going to go out 1 2 and then you're going to go down 1 2 3 4 so that's what that vector looks like that's what that vector looks like and the span of this vector is essentially all of the multiples of this vector we you know you could say linear combinations of it but there's a you're taking a combination of just one vector so it's going to be all the multiples of this vector so if I were to graph it it was just it would just be a line it would just be a line that is specified by all of the linear combinations of that vector right there this right here this right here is a graphical representation of the column space of a now let's look at the left null space of a or you could imagine the null space of the transpose they are the same thing you saw why in the last video what does this look like so the left null space is a span of 2 1 so if you graph 2 and then you go up 1 is the graph of 2 1 it looks like this let me do it in a different color so that's what the vector looks like the vector looks like that but of course we want the span of that vector so it's going to be all of the combinations or all of the in all you can do when you combine one vector is just multiply by a bunch of scaler so it's going to be all of the scalar multiples of that vector so let me draw it like that it's going to be like that and the first thing you might notice let me write this this is our left null space of a or the null space of our transpose this is equal to the left null space null space of a this is equal to the left null space of a and actually since we're writing we wrote this in terms of a transpose it's the null space of a transpose which is the left null space of a let's write the column space of a also in terms of a transpose this is equal to the row space row space of a transpose of a transpose right if you're looking at the columns of a everything it spans the columns of a are the same things as the rows of a transpose but the first thing that you see when I just at least visually drew it like this is that these two spaces look to be orthogonal to each other you know it looks like I drew it an r2 it looks like there's a 90-degree angle there and if we wanted to verify it all we have to do is take the dot product well any vector any vector in any vector that is in all in our column space so any vector you know you could pick an arbitrary vector and that's in our column space it's going to be equal to C times 2 minus 4 so let me write that down so any vector here so if V is a member let me I want to I want this stuff up here I'll scroll down a little bit so if V let's say V 1 is a member of our column space is our column space and that means that V 1 is going to be equal to some scalar multiple x times the spanning vector of our column space so some scalar multiple of this so we could say it's equal to c1 times 2 minus 4 that's some member of our column space now if we want some member of our null of our left null space right let's write it here so let's say that V 2 is some member of our left null space V 2 is some member of our left null space or the null face of the transpose now what does that mean that means v2 is going to be equal to some scalar multiple of the spanning vector of our left null space of two one of two one so any vector that's in our column space can be represented this way any vector in our left null space can be represented this way now what happens if you take the dot product of these two characters so let me do it down here I want to save some space for what we're going to do in r3 but let me take the dot product of these two characters so v1 dot v2 is equal to now arbitrarily switch colors c1 times 2 minus 4 dot C 2 times 2 1 and then the scalars we've seen this before you can just you can just say that this is the same thing as C 1 C 2 times the dot product of 2 minus 4.2 1 and then what is this equal to this is going to be equal to C 1 C 2 times 2 times 2 is 4 and then plus minus 4 times 1 minus 4 well this is going to be equal to 0 so this whole expression is going to be equal to 0 and this was for any two vectors that are members of our column space in our left null space they are orthogonal to each other so every member of our column space is going to be orthogonal to every member of our left null space or every member of the null space of our transpose and that was the case in this example it actually turns out this is going to be all this is always going to be the case that your column space of a matrix its orthogonal complement is the left null space or the null space of its transpose I'll prove that probably in the next video either the next video or the video after that but you can see it visually for this example now let's draw the other tube let's draw the other two characters that we're dealing with here so we have our we have our null space which is a span of these two vectors in r3 it's a little bit more difficult to draw it these two vectors in r3 right there but what is the span of two vectors in r3 all of the linear combinations of two vectors in r3 is going to be a plane in r3 so I'll draw it in just very general terms right here if we draw it in just very general terms and let me see so let's say it's a plane in r3 it's a plane in r3 that looks like that maybe I'll fill in the plane a little bit give you some sense of what it looks like this is this is the null space of a it's spanned by these two vectors you could imagine these two vectors look something like I'm drawing it very general but if you take any linear combinations of these two guys you're going to get stuff any vector that's along this plane that goes infinite directions and of course the origin will be in these all of these are valid subspaces now what does what does the row space what does the row space of a look like or you could say the column space of a transpose well it's all of its the span of this vector in r3 but let's let me show you something well let's just see something interesting about this vector in r3 how does it relate to these two vectors well you may not see it immediately although if you kind of look at it closely it might pop out at you that this guy is orthogonal to both of these guys notice if you take the dot product of two minus one minus three and you dotted it with one half one zero what are you going to get you're going to get 2 times 1/2 which is 1 plus minus 1 times 1 which is minus 1 plus minus 3 times zero which is zero so that's when I dotted that guy with that guy right there and then when I take the dot of this guy with that guy what do you get you get three halves zero and one dotted with let me scroll down a little bit don't want to write too small dotted with dotted with one dotted with two minus one minus three in the column so the row space of a I wrote the spanning vector there spanning vector there this time I probably shouldn't switch the order but here I'm dotting it with this guy and then here I'm dotting it with this guy right there so if you take it three has at times 2 is equal to 3 plus 0 times minus 1 is 0 plus 1 times minus 3 is minus 3 so it's equal to 0 so the fact that this guy is orthogonal to both of that both of these spanning vectors it also means that it's orthogonal to any linear combination of those guys maybe it might be useful for you to you see that so let's say that let's take some member of our let's take some member of our null space so some member of our null space so let's say that V 3 let's say the vector V 3 is a member of our null space is a member of our null space that means it's a linear combination to the linear combination of that guy and that guy those are the two spanning vectors I had written it up here these are our two spanning vectors but I need the space down here so let me scroll down a little bit these are the two spanning vectors so that means that V 3 can be written as some linear combination of these two guys that I squared off in pink so we could let me just write it as maybe a times a times 3 halves 0 1 plus B times 2 B times 1 half 1 0 now what happens if I take the dot product of v3 if I take the dot product with v3 and i dot it with any member with any member of my row space right here so any member of my row space is going to be a multiple is going to be a multiple of this guy right here that is a spanning vector of my row space so let me actually create that so let me say that V 4 V 4 is a member of my row space which is the column space of the transpose of a and that means that V 4 is equal to let's say some some scaling vectorize you see don't see a lot let me use D let's say it's d times my spanning vector d times 2 minus 1 3 so what is V 3 which is just any member of my null space any member of my null space dotted with V 4 which is any member any member of my row space so what is this going to be equal to this is going to be equal to it's going to be equal to this guy so let me write it like this a times three-halves 0 1 plus b times 1 half 1 0 dotted with dotted with this guy dot d times 2 minus 1 3 and what is this going to be equal to well we know all of the properties of vector dot products we can distribute it and then take the scalars out so this is going to be equal to I'll skip a few steps here but it's going to be equal to a d times the dot product of three-halves 0 1 dot 2 minus 1 3 just distribute it out to here plus BD plus BD times the dot product of 1 half 1 0 dotted with 2 minus 1 3 this is a dot product I just distributed this term on these two terms right here and we already know what these dot products are equal to we did it right here this dot product right here is that dot product I just switched the order so this is equal to 0 and this dot product is that dot product so this is also equal to 0 so you take any member you take any member of your row space and you dot it with any member of your null space and you're going to get you're going to get 0 or any member of your row space is orthogonal to any member of your null space and I did all of that to help our visualization so we just saw that any member of our row space which is the span of this vector is orthogonal to any member of our any member of our null space so my row space which is just going to be a line in r3 because it's just a multiple of a vector is going to look like this it's going to look like this it's going to be a line and then it's going to maybe go behind it you can't see it there it's going look like that but it's going to be orthogonal so let me draw it like so this pink line right here in r3 that is our row space row space of a which is equal to the column space of a transpose because the rows of a are the same thing as columns of a transpose and the row space is just the space spanned by your row vectors and then this is the null space of a this is the null space of a which is a plane it's spanned by two vectors in r3 or we could also call that we could also call that the left null space left null space of a transpose and I never used this term in the last video but it's symmetric right if the null space of a transpose is the left null space of a then the null space of a is the left null space of a transpose which is an interesting interesting takeaway notice the you have here the row space of a the row space of a is orthogonal to the null space of a and here you have the row space of a transpose is orthogonal to the null space of a transpose or you could say the left null space of a is orthogonal to the column space of a or you could say the left null space of a transpose is orthogonal to the column space of a transpose so these are just very interesting takeaways in general but this is shows you an end just like I said here that look these happen to be orthogonal these also happen to be orthogonal this isn't just some strange coincidence in the next video or two I'll show you that these are actually that this base this pink space is the orthogonal complement of the null space right here which means every vector in that is orthogonal to it represents all of the vectors that are orthogonal to the null space and these two guys are orthogonal complements to each other they each represent all of the vectors that are orthogonal to the other guy in their respective spaces