Lin Alg: Visualizations of Left Nullspace and Rowspace

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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.