Sasha Patotski
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 Mar30 awarded Nice Question Dec20 awarded Constituent Dec9 awarded Caucus Nov24 comment Shortlist of problems in linear algebra @RipanSaha Thanks for the link! I have seen that. It's nice, but it's not what I am looking for. First of all, it's pretty long. Second, it's rather computational. I am looking for a short list of problems, and not for complete beginners, but for someone who already knows some basics like row reduction, but is not familiar or is not familiar enough with more abstract "coordinate-free" approach to linear algebra. Nov24 comment Shortlist of problems in linear algebra @Timbuc I agree with you and disagree at the same time. I am not looking for a problem list for complete beginners, and it doesn't mean that the problems tell you everything. It's just if you know how to solve them, it means you know enough of linear algebra. Nov24 asked Shortlist of problems in linear algebra Nov18 comment K[x]-module is k-vector space Any matrix defines a linear transformation. And backwards, if you have a linear transformation, if you fix a basis in your vector space, it will give you a matrix, matrix of this linear transformation. So linear transformations are like matrices, but by default without given basis. Nov18 comment K[x]-module is k-vector space Exactly! $x$ can go to any matrix. If you know where $x$ goes, say, to matrix $A$, can you tell where $x^2$ should then go? What about $x^3$? Nov18 answered K[x]-module is k-vector space Nov11 answered Homotopy groups of pairs and homotopy fibration of inclusions Oct21 comment Why exactly is this injective? Algebraic Topology. I repeat one of my previous comments. By definition of this map it comes from the natural inclusion $ker(d^i) \to A^i$. It is by definition. Otherwise it doesn't even make sense to talk about the exact sequence. Oct21 comment Why exactly is this injective? Algebraic Topology. In your third line you said "show the sequence is exact". This "the" indicates you know the terms of the sequence and the maps between them. So what are the maps? How do you get the sequence? Do you understand where the sequence came from? Oct21 comment Why exactly is this injective? Algebraic Topology. Where did you get the map from the third formula in your question from? By definition of this map it comes from the natural inclusion $ker(d^i) \to A^i$. Oct21 comment Why exactly is this injective? Algebraic Topology. Of course the can be a non-injective map $Ker(d^i) \to A^i$. But in your case the map $Ker(d^i) \to A^i$ you are considering is by definition just the obvious inclusion. Oct21 answered Why exactly is this injective? Algebraic Topology. Oct20 answered So why isn't $\Bbb R^n = \oplus _{n = 1}^{m}\Bbb R^n$ Oct19 comment Prove that $p_M+p_{M^\perp}=I$, where $I$ is the identity on $H$. It is direct sum of vector spaces. Do you know what that is? Oct19 answered Prove that $p_M+p_{M^\perp}=I$, where $I$ is the identity on $H$. Oct15 comment Linear Algebra - Prove $AB=BA$ Since $-\frac{1}{2}A$ and $A+B$ are each other inverses, you have $-\frac{1}{2}A\cdot (A+B)=(A+B)\cdot (-\frac{1}{2}A)$ (and both equal $I$). Now multiply both sides by $-2$. Get $A(A+B)=(A+B)A$, and then open the brackets. Oct15 answered Linear Algebra - Prove $AB=BA$