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Aug
27
comment gcd of $x$ and $2$ in $Z[x]$
What do you know about the constant term of $2 \cdot \sum_i a_i x^i + x \cdot \sum_i b_i x^i$?
Jul
31
awarded  Nice Answer
Jul
30
answered Is it true that $\frac{1}{\cosh(x) - \sinh(x)} = e^{x}$?
Jul
30
comment Modulo operations over Gaussian Integers
It's almost August now, where are the pictures? Flagging for sinful lies.
Jul
29
revised How do I generalize the derivatives / integrals from multivariable calc?
Wtf am I doing with my time
Jul
26
comment Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)
Oh right, the leading term will be $-\lambda f_0$, not $-\lambda$...
Jul
26
revised Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)
added 35 characters in body
Jul
26
comment Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)
Isn't it non-surjective for all $\lambda$ though? Or is this related to domains and boundedness?
Jul
26
revised What do sine, tan, cos actually mean?
added 218 characters in body
Jul
26
comment Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)
Ah, okay. The only things I know about spectra are from a QM course I took, and we never mentioned operator norms (in fact, we didn't distinguish eigenvalues and the spectrum at all). I'll go find that book then. :)
Jul
25
comment Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)
Not the OP, but I have a question anyways: why is the spectrum just the unit disk? Isn't $R - \lambda I$ non-invertible for all $\lambda$?
Jul
25
answered What do sine, tan, cos actually mean?
Jul
25
answered Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)
Jun
21
comment How exactly do we do Gauss elimination?
After Gauss elimination, you get a matrix in reduced row-echelon form. The way I think of it is: The upper left corner is an identity matrix, the upper right is whatever, and the bottom rows are zeroes: $\left[ \begin{smallmatrix} I & F \\ 0 & 0 \end{smallmatrix} \right]$. From there, you can permute the columns if you want.
Jun
21
comment How exactly do we do Gauss elimination?
See how the bottom row is all zeros? This means we know nothing about $z$ at all. So we'll just set it to a "free parameter", $t$, and allow that to vary. In general, you'll get one parameter per zeroed-out-row (after row-reduction, of course).
Jun
21
comment Is Σ* the same as saying L*
Usually, $\Sigma$ refers to the alphabet that you're working over, and $\ast$ means "all finite strings over". In other words, $\Sigma^\ast$ is the set of all strings in your universe. On the other hand, $L^\ast$ has a very different meaning: all strings that can be made by concatenating words in $L$.
Jun
15
comment What is the limit of the ratio of the sum of all real numbers from 0 to 2 over the sum of all real numbers from 0 to 1.
You'll get the same answer: 4. The integral from $0$ to $2$ is $2$, and the integral from $0$ to $1$ is $\frac{1}{2}$.
Jun
3
comment Is the zero vector in the definition of linear dependence arbritary?
Ah, I think I see what you were aiming at. If there's two different ways to make a vector from linear combinations, then the set is dependent. The proof of this is easy: subtract the two combinations, and you'll get a non trivial combination that gives the zero vector. :)
Jun
3
comment Is the zero vector in the definition of linear dependence arbritary?
No, if you change the definition like that, you don't get anything useful. See my edit. Lemme know if you're still stuck.
Jun
3
revised Is the zero vector in the definition of linear dependence arbritary?
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