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##### Lin Alg: Visualizations of Left Nullspace and Rowspace : Relationship between left nullspace, rowspace, column space and nullspace.

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- In the last video I had this 2 by 3 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 could also call the left null space, and the row space,
- or what's essentially the space spanned by A's rows.
- 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.
- Copy, and then let me scroll down here and paste it over
- here, and hit paste.
- Let me see if I can find our key takeaways
- 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.
- Let me copy that.
- Copy that and bring it down.
- Hit paste.
- This was our column space.
- Let me write that.
- This is the column space of A.
- It was equal to that right there.
- And now what other things do we know?
- Well, we know that the left null space was a span of 2, 1.
- 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, just like that.
- And then what was our null space?
- Our null space we figured out in the last video.
- Here it is.
- It's the span of these two R3 vectors.
- Let me copy and paste that.
- Hit 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?
- So the column space of our transpose was the span of this
- R3 vector right there, So it was this one right here.
- So let me copy and paste it.
- Copy and scroll down, and we can paste it just like that.
- OK, 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, 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 2-by-3 matrix 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 codomains.
- I'll just write them very generally right here.
- So you could imagine R3 is our domain.
- And then our codomain is going to be R2 just like that.
- 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 R2 vector.
- This is a subspace of R2.
- We could write this.
- So 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.
- We could have said this guy and that guy are multiples of
- that guy, either way.
- But the basis 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.
- So let's graph them, actually.
- So I won't be too exact, but you can imagine.
- Let's see, 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 my-- do this as
- neatly as possible.
- That's my vertical axis.
- That is my horizontal axis.
- And then, 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 one, two, and then you're going to go down one, two,
- three, four.
- So that's what that vector looks like.
- And the span of this vector is essentially all of the
- multiples of this vector, where you could say linear
- combinations of it, but you're taking a combination of just
- one vector, so it's just going to be all of the multiples of
- this vector.
- So if I were to graph it, it would just be a line that is
- specified by all of the linear combinations of that vector
- right there.
- 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, it's the graph of
- 2, 1, and 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.
- All you can do when you combine one vector is just
- multiply it by a bunch of scalars, 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 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 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.
- It looks like I drew it in 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 that is in our column space, you could
- take an arbitrary vector that's in our column space,
- it's going to be equal to c times 2 minus 4.
- So let me write that down.
- I want this stuff up here.
- I'll scroll down a little bit.
- Let's say v1 is a member of our column space.
- And that means that v1 is going to be equal to some
- scalar multiple times the spanning vector of our column
- space, so some scale or 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 left null space-- let's
- write it here.
- So let's say that v2 is some member of our left null space,
- or the null space of the transpose, then
- 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 2, 1.
- So any vector that's in our column space could 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-- I'll arbitrarily switch
- colors-- c1 times 2 minus 4 dot c2 times 2, 1.
- And then the scalars, we've seen this before.
- You can just say that this is the same thing as c1, c2 times
- the dot product of 2 minus 4 dot 2, 1.
- And then what is this equal to?
- This is going to be equal to c1, c2 times 2 times 2 is 4
- 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 and our left null space.
- They're 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 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 in the
- next video or the video after that, but you can see it
- visually for this example.
- Now let's draw the other two characters that we're
- dealing with here.
- So we have our null space, which is the 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, let me see.
- So 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 the null space of A.
- It's spanned by these two vectors.
- Now, 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
- any vector that's along this plane that goes in infinite
- directions.
- And, of course, the origin will be in these.
- All of these are valid subspaces.
- Now, what does the row space of A look like?
- Or you could say the column space of A transpose?
- Well, it's the span of this vector in R3, but let's 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
- 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 2 minus 1 minus 3,
- and you dotted it with 1/2, 1, 0, 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 0, which is 0.
- 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 3/2, 0 and 1, dotted with-- let me scroll down a
- little bit.
- I don't want to write too small-- dotted with 1, dotted
- with 2 minus 1 minus 3.
- In the row space of A, I wrote the spanning
- vector there this time.
- I probably shouldn't have switched 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, 3/2 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 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 see that.
- So let's take some member of our null space.
- So let's say the vector v3 is a member of our null space.
- That means it's a linear combination of that
- guy and that guy.
- Those are the two spanning vectors.
- I'd written it up here.
- These are our two spanning vectors.
- I need the space down here, so let me scroll
- down a little bit.
- These are the two spanning vectors.
- So that means that v3 can be written as some linear
- combination of these two guys that I squared off in pink.
- So let me just write it as maybe A times 3/2, 0, 1 plus b
- times 1/2, 1, 0.
- Now, what happens if I take the dot product of v3 and I
- dot it with any member of my row space right here?
- So any member of my row space is going to be a multiple of
- this guy right here.
- That is the spanning vector of my row space.
- Just let me actually create that.
- So let me say that v4 is a member of my row space, which
- is the column space of the transpose of A.
- And that means that v4 is equal to, let's say, some
- scaling vector.
- I always use c 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 v3, which is just any member of my null space
- dotted with v4, which is any member of my row space?
- So what is this going to be equal to?
- This is going to be equal to this guy.
- So let me write it like this.
- A times 3/2, 0, 1 plus v times 1/2, 1, 0 dotted with this
- guy, dot d times 2 minus 1, 3.
- Now, 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-- ad times the dot
- product of 3/2, 0, 1, dot 2 minus 1, 3-- just distribute
- it out to here-- plus bd times the dot product of 1/2, 1, 0,
- dotted with 2 minus 1, 3.
- This is the dot product.
- I just distributed this term along 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 of your row space and you dot it with
- any member of your null space, and 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 or
- 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.
- 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 to look like that, but it's going to be
- orthogonal.
- So let me draw it.
- So this pink line right here in R3, that is our row space
- of A, which is equal to the column space of A transpose
- because the rows of A are the same thing as the columns of A
- transposed, and the row space is just the space spanned by
- your row vectors.
- And then 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 the 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 takeaway.
- Notice that you have here 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.
- And just like I said here, that look, these happen to be
- orthogonal.
- These also happen to be orthogonal.
- And this isn't just some strange coincidence.
- In the next video or two, I'll show you that this space, this
- pink space, is the orthogonal complement of the null space
- right here, which means 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.