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Apr
9
comment Do commuting matrices share the same eigenvectors?
"but no non-identity matrix has this property." — Wrong. For every number $c$, the matrix $cI$ also has that property. For $c\ne 1$ that matrix clearly is not the identity matrix (for $c=0$ it doesn't even have the same rank).
Apr
9
comment Convex hull of set of sparse vectors?
The convex hull of sufficiently many sparse vectors might contain non-sparse vectors. Where is the problem?
Apr
9
revised A pedagogical proof that 9's can be ignored when calculating digital roots
added 11 characters in body
Apr
9
comment A pedagogical proof that 9's can be ignored when calculating digital roots
@JackM: It also may have the opposite effect of convincing the students that they won't ever understand it anyway. An elementary proof followed by "but with advanced mathematics, we could have proved it in a single line" is probably much better in building interest.
Apr
9
answered A pedagogical proof that 9's can be ignored when calculating digital roots
Apr
9
comment What three odd integers have a sum of 30?
Well, one might calculate modulo 3, in which case $1+1+1+3+3\equiv 30 (\operatorname{mod} 3)$.
Apr
9
revised Is there a name for the integral used to define square integrable functions?
made the title more descriptive
Apr
9
answered The limit of $xy/(y-x^3)$ at $(0,0)$ does not exist
Apr
7
answered Zero to the zero power - Is $0^0=1$?
Apr
6
comment What would be the join and meet of this lattice?
Thank you for the clarification; with that definition, my comment doesn't apply, therefore I deleted it.
Apr
6
comment What is the difference between natural numbers and positive integers?
We also have a perfectly good way to describe the set $\{0, 1, 2, 3, \ldots\}$, namely the non-negative integers. Therefore I do not accept argument 2.
Apr
2
revised Why can one expect that $n\cdot p$ elements complete the test?
removed the wrong assertion that independence is necessary
Apr
2
comment Why can one expect that $n\cdot p$ elements complete the test?
@Henry: You're right; I just cross-checked in Wikipedia, and the addition rule indeed does not require independence. I'll fix my answer accordingly, than you.
Apr
2
comment What is mathematical definition of a fluid?
I'd say a fluid is characterized by a scalar field $\rho(x,t)$ and a vector field $j(x,t)$ which fulfil the continuity equation $\partial\rho/\partial t + \operatorname{div} j = 0$. Anything beyond that are details of the specific fluid. But that's just my guess.
Apr
2
answered Why can one expect that $n\cdot p$ elements complete the test?
Mar
22
comment If $Q$ is orthogonal, $D$ diagonal, $QDQ^T=L^TL$, how to know $L$?
With $O=Q$, $O^TQ=Q^TQ=I$ and the same for $Q^TO$. Now, $Q\sqrt{D^{-1}}Q^T = \sqrt{A^{-1}}$, so the result is the same solution you've found.
Mar
22
comment Showing a union of null sets is again a null set
But unless the sets are disjunct, their probabilities generally do not add up, so $\le$ is IMHO the right thing to use (although in hindsight we also know that $=$ is right, too, because the r.h.s. is $0$, and a probability cannot be negative). OTOH, I see nothing guaranteeing that $P(\bigcup A_n)$ actually exists ($P(A_n)$ quite obviously is allowed not to exist, or else the whole subset construction would not be needed). But I also don't see that you actually need it.
Mar
21
comment base for finite dimensional vector space is not infinite dimensional vector space?
Remember that linear combinations only have finitely many non-zero coefficients, even in infinite-dimensional vector spaces.
Mar
21
comment If $Q$ is orthogonal, $D$ diagonal, $QDQ^T=L^TL$, how to know $L$?
The matrix is not unique. If $X$ is a matrix with $X^TX=Y$ and $O$ is an orthogonal matrix, then for $X'=OX$ we have $X'^TX' = (OX)^TOX = X^TO^TOX = X^TX = Y$. It should be obvious which orthogonal matrix you want to use in the definition of $F$.
Mar
21
comment Find equivalence classes of x ~ y : <=> x-y ∈ Z
Just find a set $X$ which contains every decimal part just once; the equivalence classes are then uniquely given as $[x]=x+\mathbb Z$ for $x\in X$.