- Let's say I define determinant to have the above meaning. How can one

derive the formula for computing determinant following just the visual/geometrical meaning?

- Let's say I have an arbitrary closed 2D polytope and I transform all of its vertices by a matrix .

Is the scaling factor of polytope's area after the transformation, i.e. ?

- Imagine I have an open 2D polytope (which clearly doesn't have any area).

How does relate with the transformed polytope ?

- Suppose there's a vector . What does say about the transformed vector ?

# A few questions regarding Determinants

### #1

Posted 16 July 2016 - 02:48 AM

### #2

Posted 16 July 2016 - 08:30 AM

Let us work in and let us pick a basis .

The general motion of a volume here is via the wedge product - totally antisymmetric product of vectors.

gives the volume in the basis given above. The absolute value is with respect to the obvious Euclidean norm - we could do something more general here, but not for now.

Now let us take some linear transformation - which we know can always be written as a matrix. As a linear operator we have

.

As we are working with n vectors in an n-dimensional space and the product is totally antisymmetric the Volume is an element of a one dimensional vector space. Any changes in this volume can always be written as the volume multiplied by some scalar.

Now lets build this linear oeperator - so etc. Then we see

This scalar is the determinant of the linear transformation A.

So lets do this for the 2d case. Let us take some linear operator that I write as a matrix

Then look at its action on an arbitary vector and feed this into the wedge product; we obtain

.

Now remember that the wedge product is totally antisymmetric - which just means antisymmetric when we only have two vectors. So and we are left with

,

which is what we wanted. You could try the same thing yourself in dimension 3. In higher the same sort of thing works.

This also then gives you a solution to question 1. You can use this as a geometric definition of the determinant of a linear transformation.

I hope that helps a little

**Edited by ajb, 16 July 2016 - 08:41 AM.**

Mathematical Ramblings.

### #3

Posted 16 July 2016 - 09:27 AM

Hello ajb,

thank you for your time and efforts, this definitely helps!

I was really hopeless since I asked this question on several forums without any success – and now I finally got a response.

Thank you.

### #4

Posted 16 July 2016 - 09:48 AM

Thank you.

You are welcome.

With your question 4. I am not sure there is some general answer. However, you might want to think about eigensystems - so solutions to

Mathematical Ramblings.

### #5

Posted 16 July 2016 - 07:59 PM

You are welcome.

With your question 4. I am not sure there is some general answer. However, you might want to think about eigensystems - so solutions to

Thanks for tip, (unfortunately) I'm familiar with eigen(vectors/values).

**But I have another determinants-related question** (which could help me "intuitize" the broader meaning of determinant)**:**

Just to state some facts: AFAIK, determinant is used for:

- To solve sets of linear equations (AFAIK this is why it was first invented - by Seki in Japan).
- To compute the volume distortion of parallelepiped (AFAIK this meaning come later - introduced by Lagrange).

A matrix can represent:

- Set of linear equations (row-wise).
- The basis vectors of a coordinate system (column-wise).

*The question*: how does the two meanings of matrix (row and column) relate?

I.e. let the matrix represent a set of linear equations. What coordinate system does the matrix represent (i.e. what is the meaning of matrix **column-wise**)?

Let's say I know the above meaning. Then I guess I could connect the "Seki" and "Lagrange" meanings of determinant - for which I don't see any connection now.

I.e. I could solve a set of linear equations (represented by a matrix) graphicaly using the knowledge of the volume of parallelepiped formed by the matrix's basis vectors.

### #6

Posted 17 July 2016 - 07:32 AM

and so on... not sure that is any help though.

Mathematical Ramblings.

### #7

Posted 7 August 2016 - 01:01 AM

Think you mixed up a and c there, ajb.

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