09-240/Classnotes for Tuesday December 1

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  A complete set of the December 1st lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

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~In the above gallery, there is a complete copy of notes for the lecture given on December 1st by Professor Natan (in PDF format).

--- Wiki Format ---

MAT240 – December 1st

Basic Properties of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det : \mathbb M_{n \times n} \rightarrow F} :

(Note that det(EA) = det(E)·det(A) and that det(A) may be written as |A|.)

0. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \,\! \det(I) = 1}

1. $\det(E_{i,j}^{1}A)=-\det(A);|E_{i,j}^{1}|=-1$

Exchanging two rows flips the sign.

2. $\det(E_{i,c}^{2}A)=c\cdot \det(A);|E_{i,c}^{2}|=c$

These are "enough"!

3. $\det(E_{i,j,c}^{3}A)=\det(A);|E_{i,j,c}^{3}|=1$

Adding a multiple of one row to another does not change the determinant.

The determinant of any matrix can be calculated using the properties above.

Theorem:

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\det}' : \mathbb M_{n \times n} \rightarrow F} satisfies properties 0-3 above, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det' = \det}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(A) = \det'(A)}

Philosophical remark: Why not begin our inquiry with the properties above?

We must find an implied need for their use; thus, we must know whether a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det} exists first.