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 Nov 24 comment Understand it or burn Read "How to solve it" by Pòlya (even if you only remember his suggestion to ask yourself "can I solve an easier problem?" you still have something valuable). Try not to be too stubborn with problems you are stuck on: often you are taking the wrong approach, and no amount of thinking about that wrong approach will make it right. Start the problem from scratch with the rule that you must not attack it the same way as before. Nov 24 comment Epsilon-delta definition of limit on empty/singleton domain There isn't a "true definition," rather there are many different ones appropriate to different contexts. I agree that a reasonable definition of one-sided limit that applies to functions $f:[0,\infty) \to \mathbb{R}$ should say $\lim_{x \to 0^+}f(x)=0$ in the example you gave. Possibly the calculus text is assuming $f$ is defined on a neighbourhood of $c$ (then that iff result would be true). Nov 24 comment Epsilon-delta definition of limit on empty/singleton domain Yes, or just requiring $D$ has $c$ as a limit point, or that it contains $(a,c)$ for a one-sided limit -- it depends on the applications and the people the definition is being taught to. Nov 24 comment Epsilon-delta definition of limit on empty/singleton domain The standard way to avoid this is to only define limits for certain sets $D$, for example ones containing a punctured neighbourhood of $c$. There's no sensible way to define the limit as $x \to c$ of a function $f:\{c\}\to \mathbb{R}$ if you adopt the usual convention that $\lim_{x \to c} f(x)$ should be independent of $f(c)$. Nov 21 answered Tensor algebra and symmetric algebra Nov 20 comment Why does the fundamental theorem of calculus work? +1, but it would be good to have some brackets in the first sum so it was clear you're not summing that o(1) Nov 17 comment Property of minimal projective resolution You're welcome. The key thing when you are stuck like this is to go back and question your assumptions about the way to approach the problem (e.g. you wanted to use Fitting on $P_n$, but maybe $P_S$ was easier). Polya's How to Solve It is worth a read. Nov 17 answered Property of minimal projective resolution Nov 13 comment Matrix Reps of Associative algebra Of course if $B$ is associative and $A$ isn't then there's no injective algebra homomorphism $A \to B$. But nonassociative algebras may come with their own notion of matrix rep, e.g. Ado's theorem is a result guaranteeing faithful reps of finite-dimensional Lie algebras over fields of characteristic zero. Nov 12 revised Computing the basis of a subspace of matrices with the given nullspace. added 1 character in body Nov 11 answered Computing the basis of a subspace of matrices with the given nullspace. Nov 11 comment Computing the basis of a subspace of matrices with the given nullspace. I don't understand "because of the ones." You are trying to write down a basis. This means that in particular it has to be a spanning set, but this set doesn't span $H$. It can't do because $\dim H = 12$. I will write an answer in a moment. Nov 11 comment Computing the basis of a subspace of matrices with the given nullspace. This is better because the matrices now are in $H$. But it isn't a basis (the dimension is 12). Nov 10 comment Computing the basis of a subspace of matrices with the given nullspace. You're right that the first row of $A$ could be that, but you need to find all possible $A$. What you should take from your equations is that $A$ is in $H$ iff $a_{i5}=3a_{i3}-2a_{i2}$ for all $i=1,2,3$. So every entry of $A$ except $a_{15},a_{25}a_{35}$ can be chosen freely and then those elements are determined by the choices you have made. This helps you find a basis and the dimension. Nov 10 comment Computing the basis of a subspace of matrices with the given nullspace. Your matrix multiplication isn't quite right. $A$ is $3\times 5$ so $Av$, where $v$ is the given vector, is a height three column vector which you want to be zero if $A \in H$ (it's the sum of the rows of the 3x3 matrix you wrote down). That gives some equations the $a_{ij}$ satisfy. Nov 10 comment Computing the basis of a subspace of matrices with the given nullspace. Why not try finding the general form of a $3\times 5$ matrix whose nullspace contains that element? Then you might be able to write down a basis for $H$. To find the general form, let the entries of $A$ be $(a_{ij})$ and compute $A$ times the given vector so that you can see what equations the $a_{ij}$ must satisfy. Nov 9 comment What is Cauchy Schwarz in 8th grade terms? yes, it's right that zero is the only exception to what you had before. More briefly you have equality iff the two vectors are linearly dependent, but putting it that way doesn't fit the idea of making it accessible to 8th graders. Nov 9 comment How to prove Hom$_{k}(M,N)^{G}\cong$Hom$_{kG}(M,N)$ This is not just an isomorphism, but an equality: the $kG$-module homomorphisms $M \to N$ are the elements of $\hom_k(M,N)^G$. Nov 8 comment $A$-module and free $A$-modules just repeat the same construction with a set of generators of $\ker (A^{(I)} \to M)$. Nov 7 comment Let $s(n,k)$ denote the unsigned Stirling numbers of the first kind. Prove that… Yes! What about n=4 and three disjoint cycles?