Linear Algebra 14TBD: Setting the Stage for the NxN Determinant

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Can't believe one does so many Calculus courses without somebody explicitly saying what you said here: "The definition of integration is impractical so we use Fundamental Theorem" As you study proofs of formulas / methods of integration it becomes apparent. One tutor said there are no general analytic methods of integration, this was a step toward the insight, but you now complete it ... much thanks.Also, it seems to me your approach to determinants here is much more logical and less mysterious. The standard way is to present / assert the existence of the determinant as wonderful piece of magic which has all these applications / useful properties - it goes against the grain - but in your approach motivated by the singular idea of an algebraic test for linear dependence all unfolds with no mystery !This all gives a logical orientation - instead of a memorisation motivation.

Hythloday

Matrix diagonalization will work. time complexity N^2

turuus

6:20 we could use 'a', 'b', 'c' and 'd' just to represent row entries with suffixes '1', '2', '3' and '4' to indicate the column entry. The first term then becomes a_1*b_2*c_3*d_4 or 'a1b2c3d4'. The next entry could be written '-a1b2c4d3' and so on. (Not a solution but just making analysis of the pattern of terms for the determinant more systematic.)

abajabbajew

I'm not happy with what I'm seeing with cross product. I can post a maple sheet some where if you want. I did a 4x4 matrix, I don't want to leave a big algebraic mess here, so I'll just do numbers.
<w, x, y, z>
<(2/15)*sqrt(30), (1/10)*sqrt(30), (1/15)*sqrt(30), (1/30)*sqrt(30)>
<1/15, 4/5, 8/15, 4/15>
<-(26/225)*sqrt(30), (17/150)*sqrt(30), (17/225)*sqrt(30), (17/450)*sqrt(30)>

Determinant is 0. I was hoping for something else. The algebra also pops out zero. I was hoping to get something else. I think this means that all four are orthogonal no matter what that vector <w, x, y, z> is, and that doesn't make sense to me. I wouldn't be surprised if this pattern continued in higher dimensions, it would not be hard to check.

thomasolson

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