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.