Change of basis
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Coordinates with Respect to a Basis
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Change of Basis Matrix
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Invertible Change of Basis Matrix
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Transformation Matrix with Respect to a Basis
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Alternate Basis Transformation Matrix Example
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Alternate Basis Transformation Matrix Example Part 2
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Changing coordinate systems to help find a transformation matrix
Transformation Matrix with Respect to a Basis Finding the transformation matrix with respect to a non-standard basis
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- Let's say I've got some linear transformation T that is a
- mapping from Rn to Rn.
- So if this is its domain, which is just Rn, then its
- codomain is also Rn.
- If you give me some vector in our domain, let's call that
- vector x, then T will map it to some other member of Rn,
- which is also the codomain.
- So it'll map it over here.
- We could call that the mapping of T, or the mapping
- of x, or T of x.
- Since T is a linear transformation, we know that
- the mapping of x to its codomain is equivalent to x
- being multiplied by some matrix A.
- So we know that this thing right here is equal to some
- matrix A times x.
- You've seen all of this multiple, multiple times.
- Just to make sure we understand the wording
- properly, we said we've used the word that A is the-- we
- could either call it the matrix for T, or let's say
- it's the transformation matrix for T.
- Now, in the last couple of videos, we've learned that the
- same vector can be represented in different ways.
- It can be represented in different coordinate systems.
- When I just write the vector x like that, we just assume that
- it's being represented in standard coordinates, or it's
- being represented with respect to the standard basis.
- So let's be a little bit more particular.
- This A is the transformation for T only when x is
- represented in standard coordinates, or only when x is
- written in coordinates with respect to the standard basis.
- So let me write a little qualifier here.
- A is the transformation matrix for T with respect to the
- standard basis.
- I never wrote this blue part before.
- I never even said this blue part before, because the only
- coordinate system we were dealing with was the standard
- coordinate system or the coordinates with respect to
- the standard basis.
- But now we know that there are multiple coordinate systems.
- There are multiple ways to represent this vector.
- There are multiple ways to represent that vector, because
- Rn has multiple spanning bases.
- There are multiple bases that can represent Rn, and each of
- those bases can generate a coordinate system where you
- can represent any vector in Rn with coordinates with respect
- to any of those bases.
- So that last part I said was a bit of a mouthful, so let me
- make it a little bit more concrete.
- Let's say that I have some basis B that's made up of n--
- it has to be linearly independent.
- That's the definition of a basis-- of n vectors v1, v2,
- all the way to vn.
- Now, these are n linearly independent vectors.
- Each of these are members of Rn.
- So B is a basis for Rn, which is just another way of saying
- that all of these vectors are linearly independent and any
- vector in Rn can be represented as a linear
- combination of these guys, which is another way of saying
- that any vector in Rn can be represented with coordinates
- with respect to this basis right there.
- So the same vector x, I'm going to put
- the same dot here.
- When we represent it in standard coordinates, it's
- just going to be that right there, that vector x.
- But what if we want to represent it in coordinates
- with respect to this new basis?
- Well, then that same vector x will look like this.
- We would denote it by this.
- The same vector can be represented with respect to
- this basis.
- This could be some set of coordinates.
- This would be some other set of coordinates, but it's still
- representing the same basis.
- Likewise, this vector right here, that vector right there,
- is also in Rn.
- So it can be represented by some linear combination of
- these guys, or you can represent it with coordinates
- with respect to this basis.
- So that same point right there, I could represent it.
- So that point is this.
- But I could represent it with coordinates with respect to my
- basis just like that.
- So this is an interesting question.
- This should maybe bring an interesting
- question into your brain.
- If I start off with something that's in standard
- coordinates, and I apply the transformation T-- that's like
- applying this matrix A to it or multiplying that thing in
- standard coordinates times the matrix A-- I then get the
- mapping of T in standard coordinates.
- Now, what if I start off with that thing in nonstandard
- coordinates if I have coordinates with respect to
- this other basis here?
- Well, T should still map it to this guy.
- The transformation, no matter what, should always map from
- that dot to that dot.
- It shouldn't care what your coordinates are.
- So T should still map to that same exact point.
- T should still be a linear transformation.
- It can map from x to T of x, but that's the same thing as
- mapping from this kind of way of labeling x to this way of
- labeling x.
- So we could say maybe this guy right here could be some other
- matrix times this guy over here.
- So let me write this over here.
- These are just different coordinate systems. I
- shouldn't just even say maybe.
- This guy should be able to be represented.
- So if I represent the mapping of x in our codomain in
- coordinates with respect to B-- so that's what that guy is
- right there-- so if I want to represent that dot with this
- other coordinate system, coordinates with respect to
- this basis, it should be equal to the product
- of some other matrix.
- Let me call that other matrix D.
- Some other matrix D times this representation of x times the
- coordinates of x with respect to my alternate nonstandard
- coordinate system.
- I should be able to find some matrix D that does this.
- Then we would say that D is the
- transformation matrix for T.
- A assumes that you have x in terms of standard coordinates.
- Now D assumes that you have x in coordinates with respect to
- this basis, so with respect to the basis B.
- There's no reason why we shouldn't be able to do this.
- These things are just different ways of representing
- the exact same vector, the exact same
- dot in our sets here.
- So if I represent it one way, the standard way, I multiply
- by A, and I get Ax.
- If I represent it in nonstandard coordinates, I
- should be able to multiply it by some other matrix and get
- another nonstandard coordinate representation of what
- it gets mapped to.
- So let's see if we can find some relation between
- D and between A.
- So we learned a couple of videos ago that there's a
- change of basis matrix that we can generate from this basis.
- It's pretty easy to generate.
- The change of basis matrix is just a matrix whose columns
- are these basis vectors, so v1, v2-- I shouldn't put a
- comma there.
- These are just the columns-- v2 all the way to vn.
- This is an n-by-n matrix.
- Each of these guys are members of Rn and we have n of them.
- This is an n-by-n matrix where all of the columns are
- linearly independent, so we know that C is invertible.
- These are column vectors right here.
- So we know that C is invertible.
- We learned in the last two or three videos that if we have
- some vector x, and it's being represented by coordinates
- with respect to our basis B, we can just multiply that by
- C, and we'll get our vector x.
- This essentially will tell us the linear combination of
- these guys that'll get us x.
- Since C is invertible, we also saw that if we have the
- standard format for x, or the standard coordinates for x, we
- can multiply that by C inverse.
- That will get us the coordinates for x with respect
- to the basis B.
- These two things, if you just multiply both sides of this
- equation-- let me do it in a different color-- if you just
- multiply both sides of this equation by C inverse on the
- left-hand side, you're going to get this
- equation right there.
- Now given that, let's see if we can find
- some type of relation.
- Let's see what D times xB is equal to.
- So let's say if we take D times xB, so this thing right
- here should be equal to D times the representation or
- the coordinates of x with respect to the basis B.
- That's what we're claiming.
- We're saying that this guy is equal to D times the
- representation of x with respect to the coordinates
- with respect to the basis B.
- Let me write all of this down.
- I'll do it right here, because I think it's nice to have this
- graphic up here.
- So we can say that D times xB is equal to
- this thing right here.
- It's the same thing as the transformation of x
- represented in coordinates with respect to B, or in these
- nonstandard coordinates.
- So it's equal to the transformation of x
- represented in this coordinate system, represented in
- coordinates with respect to B.
- We see that right there.
- But what is the transformation of x?
- That's the same thing as A times x.
- That's kind of the standard transformation if x was
- represented in standard coordinates.
- So this is equal to x in standard coordinates
- times the matrix A.
- Then that will get us to this dot in standard coordinates,
- but then we want to convert it to these nonstandard
- coordinates just like that.
- Now, if we have this, how can we just figure out what the
- vector Ax should look like?
- What this vector should look like?
- Well, we can look at this equation right here.
- We have this.
- This is the same thing as this.
- Actually.
- we want to go the other way.
- We have this.
- We have that right there.
- That's this right there.
- We want to get just this dot represented in regular
- standard coordinates.
- So what do we do?
- We multiply it by C.
- Let me write it this way.
- If we multiply both sides of this equation times
- C, what do we get?
- We get this right here.
- Actually, no.
- I was looking at the right equation the first time.
- We have this right here, which is the same-- first intuition
- is always right.
- We have this, which is the same thing as this right here.
- So this can be rewritten.
- This thing can be rewritten as C inverse-- we
- don't have an x here.
- We have an Ax here, so C inverse times Ax.
- The vector Ax represented in these nonstandard coordinates
- is the same thing as multiplying the inverse of our
- change of basis matrix times the vector Ax.
- If I have my vector Ax and I multiply it times the inverse
- of the change of basis matrix, I will then have a
- representation of the vector Ax in my nonstandard basis.
- Now, what is the vector x equal to?
- Well, the vector x is equal to our change of basis matrix
- times x represented in these nonstandard coordinates.
- So this is going to be equal to C inverse A times x. x is
- just the same thing as C.
- x is just C times our nonstandard coordinates for x,
- just like that.
- Let me summarize it, just because I waffled a little bit
- on this point right there just because I got
- a little bit confused.
- If I start off with the nonstandard representation of
- x, or x in coordinates with respect to B, I
- multiply them times D.
- So if I start with this, I multiply them times D, I get
- to that point right there.
- So this right there is the same thing as
- this point right there.
- That point right there should be the nonstandard
- representation of the transformation of x, or the
- coordinates of the transformation of x with
- respect to B.
- Now, the transformation of x, if x is in standard
- coordinates, is just A times x.
- So this is just A times x.
- But I want to represent it in these nonstandard coordinates.
- Now, A times x in nonstandard coordinates is the same thing
- as C inverse times A times x, if you think this is the same
- thing as this.
- So if you have this and you want to represent it in
- nonstandard coordinates, you multiply it by C inverse, so
- then you'll get that representation in nonstandard
- coordinates.
- Then finally, we say look, x is the same thing as C times
- the nonstandard coordinate representation of x.
- So we can replace x with that right there.
- So the big takeaway here is that D times the coordinates
- of x with respect to the basis B is equal to C inverse A
- times C times the coordinates of x with respect
- to the basis B.
- So D must be equal to C inverse AC.
- So if D is the transformation matrix for T with respect to
- the basis B-- and let me write here-- and C is the change of
- basis matrix for B-- let me write that down, might as well
- because this is our big takeaway-- and A is the
- transformation-- I'll write it in shorthand-- matrix for T
- with respect to the standard basis, then we can say-- this
- is the big takeaway-- that D, our matrix D, is equal to C
- inverse times A times C.
- That's our big takeaway from this video, which is really
- interesting.
- I don't want you to lose this point.
- We now understand that A is just for a certain set of
- coordinates.
- But there's arbitrary different bases that we can
- use to represent Rn, so we can have different matrices that
- represent the linear transformation under different
- coordinate systems. If we want to figure out those different
- matrices for different coordinate systems, we can
- essentially just construct the change of basis matrix for the
- coordinate system we care about, and then generate our
- new transformation matrix with respect to the new basis by
- just applying this result.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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