albedo

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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 $\mathbf A$ represent a set of linear equations. What coordinate system does the matrix represent (i.e. what is the meaning of matrix $\mathbf A$ 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.
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 $P$ and I transform all of its vertices by a matrix $\mathbf{A}$. Is $\det\left(\mathbf{A}\right)$ the scaling factor of polytope's $P$ area after the transformation, i.e. $\det\left(\mathbf A\right) = \frac{P\text{'s area after transform}}{P\text{'s area before transform}}$? Imagine I have an open 2D polytope $\overline P$ (which clearly doesn't have any area). How does $\det\left(\mathbf{A}\right)$ relate with the transformed polytope $\mathbf {A}\overline P$? Suppose there's a vector $\mathbf x$. What does $\det\left(\mathbf{A}\right)$ say about the transformed vector $\mathbf {Ax}$? I'd be glad to get answer to any one of these questions. Thanks.