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Aug
15
comment Infinite dimensional Vector Spaces and Bases of Quotients
How would you do this in the finite-dimensional case? Your suggestion doesn't even work there: $(1,1)$ and $(1,-1)$ form a basis of $\mathbb R^2$ but your procedure does not produce a basis of the quotient of $\mathbb R^2$ by the $x$-axis.
Aug
14
comment Prove that $T^n$ is diagonalizable.
You need to assume that the matrix is not a scalar multiple of the identity.
Aug
14
comment Prove that $T^n$ is diagonalizable.
Did you try to see what happens in specific examples? Take your favorite non-diagonalizable $2 \times 2$ matrix, is its square diagonalizable?
Jul
27
comment Eigenvectors for normal operators and their adjoints
How do you know that $\langle v, (T^*-\overline{\lambda}Id)v\rangle = 0$ for all $v$ implies that $T^*-\overline{\lambda}Id = 0$?
Jul
23
comment Decompose a real symmetric matrix
Your argument that $A|_W$ has an eigenvector (in the second edit) only works if you know that $W$ is $A$-invariant, or in other words that the restriction of $A$ to $W$ actually maps $W$ to $W$ itself. This is true, but is something you should explicitly mention.
Jul
19
comment Linear Algebra - elimination and linear systems
Why did you stop doing elimination there? Your matrix isn't in echelon form yet.
Jul
13
comment Two isomorphisms of the space of vector fields/forms
In both diagrams, by $c$ and $d$ I assume you mean $M$ and $N$. If so, $a$ and $b$ are inverses of each other and $c$ and $e$ are the same. All you're doing is rephrasing the definitions of pullback and pushforward in terms of commutative diagrams.
Jul
11
awarded  Citizen Patrol
Jul
11
comment Picking an arbitrary $\epsilon$?
It is true that any positive number can be written in the form you describe, but that is not required here. Again, the definition of limit applies to all positive $\epsilon$, so all we are doing is applying that definition to one specific choice of a positive number. You could also apply the definition to the positive number $\frac1{\pi}$ to conclude that there exists $\delta > 0$ such that $|x-a| < \delta$ implies $|f(x)-l| < \frac1{\pi}$ for instance.
Jul
11
answered Picking an arbitrary $\epsilon$?
Jul
11
comment Why isn't an injection an iif?
@GitGud, it is certainly an abuse of notation, but whether or not it is "good" notation is irrelevant. It was claimed above that "it is simply not practiced. You can do it, but you're quite lonely then", and this is flatly untrue.
Jul
11
comment What is a short exact sequence telling me?
It tells you that $B$ is an extension of $C$ by $A$
Jul
11
comment Why isn't an injection an iif?
@drhab, actually Emin is correct. The notation $f(x)$ for a relation $f$ is very commonly used to denote the set of all elements related to $x$, or even in many circumstances just one possible element related to $x$. For instance, this is common when working with canonical relations in symplectic geometry or mathematical physics.
Jul
10
comment Can you use row and column operations interchangeably?
To add onto @Bye_World's comment, the answer to your question in this case is simply "yes you can use row and column operations interchangeably in this case because that is what Lang's Theorem explicitly asks for." This doesn't say anything about whether you can or cannot use them interchangeably in other scenarios.
Jul
10
comment Can you use row and column operations interchangeably?
The OP still hasn't made clear what Lang's end goal is, i.e. why he wants to end up with such a block-matrix, but here's a guess. Perhaps the claim is that for any linear map $\mathbb R^n \to \mathbb R^n$ there exists a basis for the first copy of $\mathbb R^n$ and a basis for the second copy relative to which the matrix of the linear map has the required block form. Using row and column operations together essentially amounts to changing bases of both the domain and codomain to obtain this, but this is far removed from the ordinary sense in which row operations are used to "reduce" a matrix.
Jul
10
comment Intersection of an Infinite Indexed Family of Sets
The OP's proof would not be acceptable in most "intro to proofs and sets" courses, certainly not in the one I'm teaching now. The claim that $1 + \frac1n \to 1$ (which itself probably requires justification) implies no number larger than $1$ will be in the intersection requires a clear justification.
Jul
7
comment Why can't we usually speak of partial derivatives if the domain is not open?
Taking a derivative involves taking a limit, and limits should (usually) only be taken over open sets so that you can "approach" the point from any which way possible.
Jul
6
comment $A,B$ matrices , prove $Bv = \Lambda v$
It would be good if you shared some thoughts on the problem.
Jun
28
comment If $f(x)$ is even, is $f'(0)=0$ always true
What do you get if you differentiate both sides of the equation defining what it means for $f$ to be even?
Jun
24
comment Are these parallel theorems from Set Theory and Linear Algebra connected through Category Theory?
Just a thought: maybe there is some connection if you view a pointed finite set as a vector space over the "field with one element".