Vectors and Matrices

[paulo84]

Hi,

I just have a question relative to matrices, mostly. Is the reason there are 4 values in a matrix because there are (at least in basic terms) 3 dimensions of space and one of time?

Like it seems kind of obvious, but for some weird reason in school they never state it explicitly in those terms.

Also, do matrices have any applications outside of space and time?

For future reference, can someone also please let me know if I've posted this in the right section, and if it's ok to ask questions like this, or should I be Googling/consulting a textbook/inferring...

 

[scottdave]

Matrices could be used to help solve a variety of linear systems - not just spatial ones. Solving for the currents and voltages in a circuit, for example. Depending on the complexity of the circuit, the size can be much larger than 4. There are other multivariable systems, such as weather and financial modeling, which matrices can be useful. You may find this of interest - http://ulaff.net/

 

[paulo84]

Matrices could be used to help solve a variety of linear systems - not just spatial ones. Solving for the currents and voltages in a circuit, for example. Depending on the complexity of the circuit, the size can be much larger than 4. There are other multivariable systems, such as weather and financial modeling, which matrices can be useful.

Relative to scalars and vectors, is there another one one-up from a vector?

 

[scottdave]

Relative to scalars and vectors, is there another one one-up from a vector?

A vector is a matrix which has 1 column - or 1 row, depending on how you are working with it. A matrix is a combination of multiple column vectors. Then there are tensors. You may find this YouTube video by Dan Fleisch helpful -

 

[paulo84]

A vector is a matrix which has 1 column - or 1 row, depending on how you are working with it. A matrix is a combination of multiple column vectors. Then there are tensors. You may find this YouTube video by Dan Fleisch helpful -

Thanks, that was very helpful.

 

[Mark44]

I just have a question relative to matrices, mostly. Is the reason there are 4 values in a matrix because there are (at least in basic terms) 3 dimensions of space and one of time?

No, there is no connection.
A matrix can have any number of values, if you include one-dimensional matrices (AKA vectors). Generally speaking, a matrix is a rectangular array of numbers. A 2 x 2 matrix has two rows and two columns, but you can have 2 x 3 matrices, 3 x 2 matrices, 8 x 8 matrices, and on and on.

Like it seems kind of obvious, but for some weird reason in school they never state it explicitly in those terms.

For good reason.

Also, do matrices have any applications outside of space and time?

See the wiki article, https://en.wikipedia.org/wiki/Matrix_(mathematics), under "Applications".

For future reference, can someone also please let me know if I've posted this in the right section, and if it's ok to ask questions like this, or should I be Googling/consulting a textbook/inferring...

This should probably go in the math technical section. For basic questions like this, it's a good idea to start by a web search to get some basic information. For more information, there are lots of linear algebra books out there that discuss matrices in much greater detail.

 

[paulo84]

Ok, makes sense if you're dealing with multiple dimensions of space and/or time, as well as the other applications mentioned.

 

[mfb]

Special relativity has many vectors with 4 entries and matrices with 4*4=16 entries because we have 3 space and 1 time dimension. The concepts of vectors and matrices are much more general.

 

[paulo84]

Special relativity has many vectors with 4 entries and matrices with 4*4=16 entries because we have 3 space and 1 time dimension. The concepts of vectors and matrices are much more general.

Could you please share a little more about the 16 entry matrices which relate to space and time?

 

[jedishrfu]

One example would be the metric tensor of General Relativity

https://en.m.wikipedia.org/wiki/Metric_tensor_(general_relativity)

It’s a 4x4 matrix of values that's used to define time, distance, volume, curvature... in a spacetime.

 

[paulo84]

One example would be the metric tensor of General Relativity

https://en.m.wikipedia.org/wiki/Metric_tensor_(general_relativity)

It’s a 4x4 matrix of values that's used to define time, distance, volume, curvature... in a spacetime.

That is extremely impressive. I'm going to have to read it a number of times.

 

[jedishrfu]

Here's a video of on the Einstein Field equations that show applications of the metric tensor and other 4x4 marices: