linear algebra - Why interchanging two rows of matrix result in . . . Why interchanging two rows of matrix results in negative determinant? After thinking it a bit I felt: interchanging rows = flipping orientation of transformation represented by matrix OR change orientation of space = -ve determinant scaling factor of transformation Q1 Is it correct? Q2 Also is this correct for all dimensions? Q3
calculus - Reversing the Order of Integration and Summation . . . The more general question is about interchanging limits and integration With infinite sums, this is a special case, because by definition $\sum_{n=1}^\infty f_n(x) = \lim_{N \to \infty} \sum_{n=1}^N f_n(x)$ So because one can always interchange finite sums and integration, the only question is about interchanging the limit and the integration
sequences and series - Interchange finite and infinite sum . . . Interchanging with a finite sum correspond to a different shuffling of the series terms It is known that terms of a conditionally convergent series (i e convergent, but not absolutely) can always be shuffled to produce any number If both sums are finite there is no trouble at all to switch them $\endgroup$ –
Why can we interchange summations? - Mathematics Stack Exchange Any nondecreasing sequence converges to its (possibly infinite) supremum Thus a series of nonnegative terms converges to the supremum of its partial sums and interchanging the order of summation doesn't affect the value of the supremum: there is no accidental cancellation of terms of opposite sign