Expansion of space

[Agent Smith]

Space expansion is an accepted theory in cosmology. Lawrence Kraus illustrates this as points on a grid. From any point on that grid, it looks as though

1. Every other point is moving away from that point
2. n times as far away, n times greater the expansion

I've been learning matrices and am familiar with basic transformations of a given point/vector as the "weighted sum"

For a \(\displaystyle 2 \times 2\) e.g. \(\displaystyle \begin{bmatrix}a & b \\ c & d \end{bmatrix}\)transformation matrix applied to the vector/point \(\displaystyle \begin{bmatrix} x \\ y \end{bmatrix}\), we have the image at:
\(\displaystyle x \begin{bmatrix} a \\ c \end{bmatrix} + y \begin{bmatrix} b \\ d \end{bmatrix}\)

What kinda transformation matrix could be used to represent the expansion of space (cosmology)? Is there another (easier) way of doing this?

 

[blamocur]

What kinda transformation matrix could be used to represent the expansion of space (cosmology)? Is there another (easier) way of doing this?

What else have you learned about matrices? E.g., can you write a matrix which multiplies every vector by 3 ?

 

[Agent Smith]

What else have you learned about matrices? E.g., can you write a matrix which multiplies every vector by 3 ?

The simplified version of the real McCoy was 2 dimensional (thanks Lawrence Krauss for trying to make us understand). I have seen rather complicated transformations of the plane in videos (baffling).

As for your question, are you referring to scalar multiplication, looks like you are. The lessons I took are introductory, a lot of intermediate steps were skipped. Dunno if I can answer your question as a yes or no.

 

[mario99]

The simplified version of the real McCoy was 2 dimensional (thanks Lawrence Krauss for trying to make us understand). I have seen rather complicated transformations of the plane in videos (baffling).

Where are the videos?

The simplified version of the real McCoy was 2 dimensional (thanks Lawrence Krauss for trying to make us understand). I have seen rather complicated transformations of the plane in videos (baffling).

As for your question, are you referring to scalar multiplication, looks like you are. The lessons I took are introductory, a lot of intermediate steps were skipped. Dunno if I can answer your question as a yes or no.

Agent Smith, I like you, but I can't understand most of your questions. The funny thing is that I am a Matrix expert and you are a Matrix beginner, but I can't understand your Matrix questions!

Let us assume that there is a Matrix that describes the expansion of Space. If you can't understand a matrix which multiplies every vector by 3, how the heck you will be able to understand the Space Matrix?

 

[Agent Smith]

Where are the videos?


Agent Smith, I like you, but I can't understand most of your questions. The funny thing is that I am a Matrix expert and you are a Matrix beginner, but I can't understand your Matrix questions!

Let us assume that there is a Matrix that describes the expansion of Space. If you can't understand a matrix which multiplies every vector by 3, how the heck you will be able to understand the Space Matrix?

I was taught (regarding transformations of the plane) that if \(\displaystyle \begin{bmatrix}1 \\ 0 \end{bmatrix} \to \begin{bmatrix}3 \\ 2 \end{bmatrix} \wedge \begin{bmatrix} 0 \\ 1 \end{bmatrix} \to \begin{bmatrix}4 \\ 5 \end{bmatrix}\) (unit vectors and their images) then the image of the point/vector \(\displaystyle \begin{bmatrix}7 \\ 8 \end{bmatrix}\) is \(\displaystyle 7 \begin{bmatrix} 3 \\ 2 \end{bmatrix} + 8 \begin{bmatrix} 4 \\ 5 \end{bmatrix} \)

Texas university had an app that showed the changes to a pic of a dog when 2 × 2 matrix transformations were applied to it. It's no longer available.

 

[Agent Smith]

If I wanted to do a reflection across the \(\displaystyle y\) axis then the transformation matrix would be \(\displaystyle [\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\). That's a start, oui? Assuming I'm correct

 

[Agent Smith]

I'm interested in space dilation and it seems for that, if I want a \(\displaystyle n \times\) dilation, the transformation matrix is \(\displaystyle \begin{bmatrix}n & 0 \\0 & n \end{bmatrix}\) for a point/vector \(\displaystyle \begin{bmatrix}a \\ b \end{bmatrix}\) and \(\displaystyle \begin{bmatrix}n & 0 \\ 0 & n \end{bmatrix} = n \begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix}\). Does that mean dilation is effected by multiplication by a scalar?

 

[Agent Smith]

However the expansion of space dilation has another feature: take 2 points, A and B . If B is n times farther away than A (from the origin), B undergoes a dilation that's n times the dilation A undergoes. How do we encode this into a transformation matrix?

 

[blamocur]

As for your question, are you referring to scalar multiplication, looks like you are.

Yes, I am. Scalar multiplication can be represented by a matrix -- do you know how it looks?

 

[blamocur]

I'm interested in space dilation and it seems for that, if I want a \(\displaystyle n \times\) dilation, the transformation matrix is \(\displaystyle \begin{bmatrix}n & 0 \\0 & n \end{bmatrix}\) for a point/vector \(\displaystyle \begin{bmatrix}a \\ b \end{bmatrix}\) and \(\displaystyle \begin{bmatrix}n & 0 \\ 0 & n \end{bmatrix} = n \begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix}\). Does that mean dilation is effected by multiplication by a scalar?

Looks good.

Does that mean dilation is effected by multiplication by a scalar?

Not sure I understand this question, but linear dilation is multiplication by a scalar, or by an equivalent matrix which you've figured out earlier.

 

[blamocur]

However the expansion of space dilation has another feature: take 2 points, A and B . If B is n times farther away than A (from the origin), B undergoes a dilation that's n times the dilation A undergoes. How do we encode this into a transformation matrix?

Unless I misunderstood this sounds like a non-linear dilation, which means it cannot be represented by a matrix -- or scalar -- multiplication.

 

[Agent Smith]

Unless I misunderstood this sounds like a non-linear dilation, which means it cannot be represented by a matrix -- or scalar -- multiplication.

It's confusing. What does that statement actually mean?

For simplicity let's just stick to a 1D number line. There's a point 1 and a point 2 (the numbers 1 and 2). 2 is twice further from 0 than 1. I apply a \(\displaystyle 2 \times\) dilation. 1' = 2 and 2' = 4. So Did 2 undergo \(\displaystyle 2 \times\) the dilation of 1? No it didn't, right? If it did, 2' = (2 × 2) × 2 = 8. Do you mean to say we can't have a transformation matrix for the above, because it's nonlinear? Then how did Lawrence Kraus manage to create a CGI of space expansion in 2 D?

 

[Agent Smith]

The videos are on Khan Academy.

 

[Agent Smith]

I suppose this is relative dilation. What happens to a point is relative to what happens to other points. Could we do this: Say we know the relative positions of 2 points A and B from the origin. B is n times further away than A. For any given dilation (k) for A, we could apply the dilation nk to the point B and all other points that are n times further away. We would need to compute the relative distances then for all points and adjust the dilator (nk). Je ne sais pas.

 

[blamocur]

There's a point 1 and a point 2 (the numbers 1 and 2). 2 is twice further from 0 than 1. I apply a 2×\displaystyle 2 \times2× dilation. 1' = 2 and 2' = 4. So Did 2 undergo 2×\displaystyle 2 \times2× the dilation of 1? No it didn't, right? If it did, 2' = (2 × 2) × 2 = 8.

This is the way I understood your post #8, i.e. that '2' would dilate to '8'. But it seems that you meant constant dilation/scale factor, i.e. a linear transformation.

 

[Agent Smith]

This is the way I understood your post #8, i.e. that '2' would dilate to '8'. But it seems that you meant constant dilation/scale factor, i.e. a linear transformation.

Ok, gracias