albedo

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 [latex]\mathbf A[/latex] represent a set of linear equations. What coordinate system does the matrix represent (i.e. what is the meaning of matrix [latex]\mathbf A[/latex] 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.

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.

Hi, I know how to compute determinants and I'm familiar with the geometrical meaning of determinant as the scaling factor of a unit (point/square/cube/hypercube)'s area/volume by applying a linear transformation (using a matrix). However, I have several questions: 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 [latex]P[/latex] and I transform all of its vertices by a matrix [latex]\mathbf{A}[/latex]. Is [latex]\det\left(\mathbf{A}\right)[/latex] the scaling factor of polytope's [latex]P[/latex] area after the transformation, i.e. [latex]\det\left(\mathbf A\right) = \frac{P\text{'s area after transform}}{P\text{'s area before transform}}[/latex]? Imagine I have an open 2D polytope [latex]\overline P[/latex] (which clearly doesn't have any area). How does [latex]\det\left(\mathbf{A}\right)[/latex] relate with the transformed polytope [latex]\mathbf {A}\overline P[/latex]? Suppose there's a vector [latex]\mathbf x[/latex]. What does [latex]\det\left(\mathbf{A}\right)[/latex] say about the transformed vector [latex]\mathbf {Ax}[/latex]? I'd be glad to get answer to any one of these questions. Thanks.