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Determinant as volume of the transformation is cool, but I still don't "get" the determinant intuitively despite many years of math. In particular, why does the
by kens 14y ago
Determinant as volume of the transformation is cool, but I still don't "get" the determinant intuitively despite many years of math. In particular, why does the determinant work to solve linear equations (i.e. Cramer's rule)? And what's the motivation behind the formula for the determinant? (I realize these questions are a bit vague, but I'm hoping for a more intuitive answer than "that's just the way the math works out".)
- jules 14y agoOne approach to this is geometric algebra. I couldn't find a good reference that explains it intuitively, but there's this: http://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra#Matrix_Related http://en.wikipedia.org/wiki/Comparison_of_vector_algebra_an...
- vjk2005 14y agoreg. "And what's the motivation behind the formula for the determinant?", if I'm understanding your question right, the neat little pic at http://en.wikipedia.org/wiki/Determinant#2-by-2_matrices http://en.wikipedia.org/wiki/Determinant#2-by-2_matrices - might be what you were looking for.
- adeelk 14y agoAs is often the case, the way to get the intuition here is to just work out the computation. Once you do that, it should all "click".
- jimhefferon 14y ago> And what's the motivation behind the formula for the determinant? FWIW, that book has two explanations: the first on p 296 is a lot like "that's how it works out" and the second on p 320 is geometric. > why does the determinant work to solve linear equations (i.e. Cramer's rule)? Does the explanation on p 331 of joshua.smcvt.edu/linearalgebra/book.pdf help? (It uses the geometric understanding of the determinant.)