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seen Jun 23 at 13:02

Jun
27
awarded  Nice Answer
May
25
comment An equation that generates a beautiful or unique shape for motivating students in mathematics
There ain't any cooler equation than this one ;-)
May
25
answered Is the operator $A$ self-adjoint? unitary? normal?
May
24
awarded  Nice Answer
May
21
comment If $F$ is a closed nowhere dense subset of $\mathbb{R}$, and I define $f_n(x) = \frac{1}{n}$ for $x \in F$, is $f_n(x)$ continuous?
Continuous on what set? If $f_n$ is supposed to be a map from $\mathbb{R}$ to $\mathbb{R}$, then how is $f_n$ defined for $x \notin F$? Is $f_n$ zero there? If you view $f_n$ as a map from $F$ to $\mathbb{R}$, then $f$ is certainly continuous because it is constant.
May
21
comment Can you find a ellipse so that its image is a circle?
@mesel Ah, OK, I think I see what you want. You want a projection onto $P$ in the $z$-direction, i.e. You map $(x,y,0)$ to $(x,y,1-x-y)$, right?
May
21
comment Can you find a ellipse so that its image is a circle?
@mesel Yes, but what are the constraints on this transform $F$? You can, for example, find a rotation+translation that maps the $xy$-plane into $P$, and the problem is trivial then, since rotations and translations map circles to circles. But that's quite certainly not what you want. The minimal requirement you probably want to impose is probably that $F$ is idempotent, meaning that applying it twice is the same as applying it once. That makes $F$ a projection. But there are still many such projections (because you can pick the direction the light comes from)...
May
21
answered Does a System to Automatically (Mathematically) Generate Arbitrary Computer Programs Really Exist?
May
21
comment Can you find a ellipse so that its image is a circle?
Does "shadow" mean it's orthogonal projection to $P$? Or any projection?
May
18
comment matrix of orthogonal projection with respect to the ordered basis.
Look at Christiaan Hattingh's answer for how to convert from your basis to the canonical basis.
May
18
comment matrix of orthogonal projection with respect to the ordered basis.
BTW, using MathJax instead of ascii art to typeset formulas would have made your question much more readable. So please use MathJax in your next question...
May
18
comment matrix of orthogonal projection with respect to the ordered basis.
Your solution is correct. Remember that the matrix you found is the matrix representation of $T$ with respect to the basis $B = \{(1,2),(-2,1)\}$, not with respect to the canonical basis. So sending $(1,0)_B$ to $(1,0)_B$ and $(0,1)_B$ to $(0,0)_B$ is what $T$ was designed to do - if you translate back to the canonical basis, $(1,0)_B$ corresponds to $(1,2)$, and $(0,1)$ to $(-2,1)$, so in canonical coordinates $T$ indeed sends $(1,2)$ to $(1,2)$ and $(-2,1)$ to $(0,0)$.
May
17
comment This function is injective
@user42912 Because all the summands are positive, and thus if the sum converges at all (which it does), it converges absolutely. That suffices to guarantee that the order is irrelevant.
May
17
comment This function is injective
@MartinArgerami That's not true, strictly speaking. The sum could diverge, or the sum's value could depend on the summation order.
May
16
comment Why can't you add apples and oranges, but you can multiply and divide them?
@ColeJohnson Well, it can make sense to divide apples and oranges. Take, for example "This fruit punch contains two apples per orange" - in other words, it's recipe is $2$ apples/orange. This is also orthogonal to the original question - scale invariance prevents us from adding quanties with different units, but not from multiplying or dividing them. As I said, scale invariance is not something that we'd expect of apples and oranges, so we may, in fact, add them. How we interpret diving them has nothing to do with that...
May
16
comment Complete Normed Space => Uncountable Hamel basis not by Baire
There's a hole in that argument, I think. You seem to be arguing that no finite sum of basis vectors can yield $x$, because $x$ is defined as an infinite sum of such vectors. But how do you know that? Linear independence only requires finite linear combinations to be unique, infinite linear combinations can pretty much behave however they want, I'd say.
May
15
comment What is max norm of a matrix?
Yup. It's that simple ;-)
May
15
comment What is max norm of a matrix?
The maximum norm of $A$ is $8$ in this case.
May
15
comment Question on expectation and mean value
@ClementC. The last part of your comment is a bit missleading. A random variable can take arbitrarily large values and have finite expectation. But if the probability of taking larger and larger values doesn't drop off "fast enough", then the expectation is infinite, even though all the values individually are finite. Saying "... and even if this happens with very small probability this is enough to make the expectation infinite" OTOH gives the impression that all unbounded random variables have infinite expectation.
May
15
answered How to notate a vector out of a $\mathbb{R}^+$