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Jan
22
awarded  Notable Question
Jan
18
answered Why does a matrix with zero eigenvalues and nonzero singular values have to be non-symmetric?
Jan
17
comment Less suggestive terms for “vector addition” and “scalar multiplication”
It's also quite uncommon to denote a maximum with $\min$ ☺. I guess you wanted to write "minimum" here.
Jan
17
revised How to compute $\cos(\arctan(2)) = 1/\sqrt{5}$
fixed math formatting and spelling
Jan
17
comment Let $2 = \{ 0, 1 \}$ and $X$, $Y$ be sets.
@GitGud: I disagree. The best definition of 2 is still "the successor of 1". Of course if you model the natural numbers with the von-Neumann construction, the successor of 1 turns out to be $\{0,1\}$. But that's already a result, not the definition.
Jan
17
revised Taylor expansion of $\cosh^2(x)$
fixed MathJax formatting
Jan
17
comment Function with infinitely many right inverses?
Indeed, with that method you can easily show that the set of right inverses can have any cardinality, by just replacing $\mathbb N$ with a set of the desired cardinality.
Jan
17
comment a vectorspace, a linear map, the kernel and image of it
@Bernard: I don't think the Babylonians defined whether $0$ belongs to the set of natural numbers. Peano defined the natural numbers both without and with $0$, and used the notation $N_0$ for the latter. So there's precedent for that notation as old as the axiomatization of natural numbers themselves.
Jan
17
comment If the sets $A=\{x\in E, f(x)<\lambda\}$ and $B=\{x\in E, f(x)>\lambda\}$ are open, then $f$ is continuous
Strictly speaking, you haven't covered open sets like $(a,b)\cup (c,d)$. OTOH, the empty set is already covered by $(-\infty,b)\cap(a,\infty)$ through the case $a\ge b$.
Jan
17
comment Proving that basis always exists and is not unique
Of course to prove non-uniqueness for dimension $>1$ (the precondition to having two basis elements $e_1,e_2$ to begin with), you can also just replace $e_1$ with $e_1+e_2$. That works even for $\mathbb F_2$.
Jan
16
comment Is there a relationship between isometry as defined on metric spaces and those on vector spaces?
"I look up the definition of isometry online and many sources tell me it is a bijective "structure preserving" map." That sounds more like the definition of an isomorphism. Now an isometry is an isomorphism, but in general an isomorphism is not an isometry.
Jan
16
answered Probability of urns
Jan
16
revised Any subspace of connected set is connected?
added 5 characters in body
Jan
10
comment In Wikipedia's motivating example of wedge product, what happened to $e_1 \wedge e_1$
@Bye_World: So "alternating" does not mean the same as "antisymmetric"? I thought the property you seem to call "alternating" is called "nilpotent".
Jan
8
answered In Wikipedia's motivating example of wedge product, what happened to $e_1 \wedge e_1$
Jan
8
comment Counting how many possible images in an 800x500(3color) image.
Yes, as I wrote above: The most significant byte would be 1, and the 1,200,000 other bytes would be 0. However note that you only need 1,2000,000 bytes to represent $256^{800*500*3}$ different values (which would, obviously, not include the value $256^{800*500*3}$; the highest representable value in 1,200,000 bytes is $256^{800*500*3}-1$).
Jan
8
comment Counting how many possible images in an 800x500(3color) image.
Looking again at the three-digit example, interpreting the digits as number gives numbers from $0$ to $999$. That are $1000$ numbers, but the number $1000$ itself is not among them; indeed, it's the first one that's too large to fit in three digits. Another way to see it is: If it were only the numbers $1$ to $999$, you'd have $999$ numbers. But additionally you've got the $0$, and thus one more. But $999$ is already the largest 3-digit number, so you get a four-digit number. BTW, to write a positive integer $n$ in base $b$, you always need $\lfloor\log n/\log b\rfloor + 1$ digits.
Jan
8
comment Counting how many possible images in an 800x500(3color) image.
Well, that expression indeed gives exactly $1\,200\,000$. However, consider the number of sequences of 3 decimal digits. There are $10^3=1000$ such sequences, but the number $1000$ already has 4 digits (but $\ln 1000/\ln 10=3$).
Jan
8
comment Counting how many possible images in an 800x500(3color) image.
Actually, you need $1\,200\,001$ bytes to represent that number: The first byte is $1$, all others are $0$.
Jan
8
asked Functions where the pre-image of convex sets is convex