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10h
comment How to Find the pointwise limit of $(f_n)$
if it's point-wise you can just use l'hopital's rule (differentiating with respect to $n$)
11h
comment For what values the real parameter t is a matrix diagonalisable?
You need to guarantee that each eigenvalue's algebraic and geometric multiplicities are equal. So if $t=1$, then $\lambda=1$ better have three eigenvectors associated to it, if $t\neq1$, then 2.
22h
comment What is Bootstrapping in statistics? How can I use it to determine error in the mean, variance, kurtosis and skewness of a data set?
you might want to post this instead on the statistics stack exchange. Briefly, you treat the sample data as the population and sample with replacement using your new sample to calculate the confidence interval of some statistic in order to determine the robustness of your estimate of said statistic.
22h
comment relation between trace of product and sum of matrices?
for symmetric matrices, $tr(AB)=\sum_{i,j}a_{ij}b_{ij}$, and $tr(A+B)=\sum_{i}a_{ii}+b_{ii}$. I'm thinking you could construct a counter example to either inequality direction.
23h
comment Find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$
you're welcome!
23h
comment Find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$
having distinct eigenvalues is actually a stronger condition than non-defectivity. A matrix is non-defective if for each eigenvalue its algebraic multiplicity is equal to its geometric multiplicity. Or conversely, a matrix is defective if it has an eigenvalue whose geometric multiplicity is less than its algebraic multiplicity, which means its eigenvectors don't form a basis for the vector space.
23h
comment Find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$
as long as the matrix is non-defective (distinct eigenvalues is enough), then the eigenvalues of $A^n$ are just the eigenvalues of $A$ all raised to the $nth$ power. Raising the eigen-decomposition to the $nth$ power and simplifying shows you why.
23h
comment Finding the characteristic polynomial of $A^2$ given the characteristic polynomial of $A$
try an eigen-decomposition
23h
comment Find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$
try an eigen-decomposition
1d
comment Differences between real and complex analysis?
I would say your impressions are not correct, sequences, series, derivatives, integrals, limits, are the fundamental building blocks of analysis and to my mind real and complex analysis deal with them about equally. I find a lot of real analysis makes critical use of the fact that $\mathbb{R}$ is totally ordered, $\mathbb{C}$ loses order but gains a lot of 'elegant' properties such as well behaved derivatives, algebraic closure, etc.
1d
comment A question related to measura space
@Aera, what part can't you understand?
1d
comment Intuition behind independence result
But remember, the number of ways to obtain a large $X$ is much higher when the sample size is significantly bigger than $X$, than if it was something like $N=X+1$, (consider $\binom{X+1}{X}$ vs $\binom{2X}{X}$) so this counter-balances the fact that for any particular $N$ a larger $X$ means $Y$ must be smaller.
1d
comment Geometry and Algebra, Algebraic Geometry Question
yes, geometry, yes, $I(V(J))=\sqrt{J}$
1d
comment Why is the implication “If pigs could fly, I'd be king” a true implication?
it might be more intuitive to think about it in disjunctive form, one of these two statements must be true: either 'pigs can't fly' or 'I'm king'.
1d
answered A question related to measura space
1d
comment Show that $f$ is constant in $D(0,1)$.
is $D(0,1)$ the unit disk centered at the origin? and how can $|f(0^2)|$ be strictly greater than $|f(0)|$?
May
23
comment How do I calculate $ \int_{-\infty}^{\infty} e^{-ax^2} \;\mathrm{d}x$ for $a>0 $
this is a famous integral with no anti-derivative representable in terms of special functions, you could probably just google it
May
11
comment How can I evaluate the following limit-integral combination?
@abiessu, no it doesn't, plus taking the limit of the dummy variable wouldn't make sense anyways.
May
11
comment How can I evaluate the following limit-integral combination?
@abiessu, it's x, I checked it numerically.
May
4
comment Find a basis and the dimension of the eigenspaces of the matrix
This is a symmetric matrix, hence its eigenspace has dimension 3 and you can just choose the standard basis for $\mathbb{R}^3$, no need to find the eigenvectors.