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Apr
18
comment Proof that $\{n\}$ is a Cauchy Sequence. Where is the fallacy?
That gives $-n < \epsilon/2$, not $n < \epsilon/2$.
Apr
18
comment Proof that $\{n\}$ is a Cauchy Sequence. Where is the fallacy?
How do you go from $n > -\epsilon/2$ to $n < \epsilon/2$?
Apr
10
comment How can two equations with three variables be in R2, not in R3?
Their intersection does not lie in $\mathbb R^2$.
Apr
10
comment How can two equations with three variables be in R2, not in R3?
It makes no sense to say a $2 \times 3$ matrix is in $\mathbb R^2$. Can you please clarify what you think this means or where you heard this?
Apr
5
comment Setting up a double integral over a rectangle
The notation $[1,2] \times [3,4]$ refers to the rectangle consisting of points where the $x$ coordinate is in the interval $[1,2]$ and $y$ coordinate in the interval $[3,4]$, so $1 \le x \le 2$ and $3 \le y \le 4$.
Mar
30
comment Can $S_n$ be a cyclic group?
Any cyclic group is abelian, and very few $S_n$ are abelian.
Mar
18
comment Prove that two polynomials of degree $m$ and $n$ intersect in at most $m$ points
What does it mean for polynomials to intersect? Do you mean that when viewing them as functions, their graphs intersect?
Mar
8
comment How to show matrix $A$ diagonalizable iff $A^k$ is diagonalizable for $k\ge 2$?
The claim in the title is not true.
Mar
8
comment Unitary Matrix columns = Orthonormal Basis
What definition of "unitary matrix" are you using?
Feb
6
comment How to find the tangent space of a general submanifold?
$T_pS$ is the subspace consisting of all $X \in T_pM$ such that whenever $f$ is a function which vanishes on $S$, then $Xf = 0$.
Feb
5
comment If $f$ is differentible at a point $x \in [a,b]$, then $f$ is continuous at $x$.
If $f(t)-f(x) \to 0$ as $t \to x$, then $\lim_{t \to x} f(t)=f(x)$, which is one way of phrasing what it means for $f$ to be continuous at $x$.
Feb
5
comment Which metric is used in this limit
Yes, the metric on $\mathbb R$ is used to say that the sequence $\operatorname{diam} E_n$ of real numbers converges to $0$ is the standard absolute value metric.
Jan
28
comment Linear systems of equations and vector spaces
Why did you delete half a body of text? The answer below no longer makes sense.
Jan
22
comment Derivative of linear transformation with confusing moment
In 2), you're misinterpreting $A'(x_0)$. This is NOT a number but rather a $1 \times 1$ matrix whose only entry is $2$, with the point being that $A'(x_0)$ is meant to be a linear transformation. The linear transformation determined by the $1 \times 1$ matrix $A'(x_0)$ is the same as that determined by $A$.
Dec
14
comment Total confusion about differential one-forms and non-coordinate bases
There is no phantom $e_k$ on the right-side of the equation you speak of: you must read $e_k(f)$ as ONE expression, which is the tangent vector $e_k$ evaluated at $f$. Recall that a tangent vector in the end is a map from functions to numbers, so $e_k(f)$ is a number.
Dec
7
comment Where am I going on wrong on this integral?
Check your region: your proposed integral would correspond to inner limits of $\int_y^5$ in the original integral.
Dec
5
comment Multiple eigenvectors for an eigenvalue and how to know
To clarify, any eigenvalue has infinitely many eigenvectors. What you are asking about is the maximum number of linearly independent eigenvectors it can have.
Nov
30
comment How many n derivatives do you take for Taylor series to be accurate?
How are you defining "accurate"?
Nov
13
comment Can i use complex analysis to solve a vector calculus problem?
The fundamental reason why there is a difference is that multiplication of complex numbers makes sense, whereas multiplication of points of the $xy$-plane does not. This leads to a definition for complex differentiability analogous to that of differentiability for a function $\mathbb R \to \mathbb R$, which is not possible if you think of $\mathbb C$ as simply $\mathbb R^2$.
Oct
16
comment How are groups with the same Lie Algebra inequivalent?
What definition of "equivalent" are you thinking of?