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May
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Mar
16
comment Curve in a product of tori
Do you know a criterion for denseness when, say, $n = 2$?
Mar
9
comment Elementary set problem
@RobertCardona: It implies that $A = \{f^n(x)\}_{n=1}^\infty$, which in particular means that $x = f^n(x)$ for some $n \geq 1$, and this cyclicity implies that $A$ is finite.
Mar
5
answered Quick group question on $GL(n)$
Mar
5
comment Quick group question on $GL(n)$
Matrices over $\mathbb{R}$, you mean?
Feb
9
comment How many binary sequences of length 7 have at least two 1's?
What do you get if you count those that have exactly zero or exactly one 1?
Feb
5
comment How to prove that the minimum parameters required is equal to the dimension of an object?
And maybe I should say that as soon as you understand the various concepts involved in the statement, the case $n = 1$, $m = 2$ isn't too bad; it's probably on Stack Exchange somewhere.
Feb
5
comment How to prove that the minimum parameters required is equal to the dimension of an object?
[...] out with understanding the notion of a homeomorphism though. Let me once again stress that this is a story about topological manifolds, and that depending on your viewpoint, other answers would apply.
Feb
5
comment How to prove that the minimum parameters required is equal to the dimension of an object?
[...] turns out to be impossible; the precise and general statement is the following: there is no continuous bijective map $f : \mathbb R^n \to \mathbb R^m$ with a continuous inverse $f^{-1} : \mathbb R^m \to \mathbb R^n$. This follows from the so-called invariance of domain theorem, which it actually requires a fair amount of mathematical machinery to prove. Again, you can see the Wikipedia article for some discussion. If you're not already familiar with it, I would suggest starting [...]
Feb
5
comment How to prove that the minimum parameters required is equal to the dimension of an object?
Good question! I kind of swept that under the carpet by saying that if there is an $n$ as in my answer, then it's uniquely determined. So, let's take for granted that the solid square is $2$-dimensional. That is, every point has coordinates described by pairs of numbers. Would one number do? If so, by the definition above, a given point would have neighbourhoods with weird properties: one that could be continuously deformed to $\mathbb R$ and to $\mathbb R^2$ (in a one-to-one fashion). In particular, this would imply that $\mathbb R$ itself could be transformed to $\mathbb R^2$. This [...]
Feb
5
answered How to prove that the minimum parameters required is equal to the dimension of an object?
Feb
5
comment How to prove that the minimum parameters required is equal to the dimension of an object?
It seems you're essentially answering your own question then, by defining the dimension in terms of the number of parameters needed, at least insofar that you view the parameters as defining coordinates. I'll elaborate a bit in an answer.
Feb
5
comment How to prove that the minimum parameters required is equal to the dimension of an object?
Do you have a favourite definition of "dimension", or "object" for that matter? As you might imagine, the answer to your question will depend on that.
Feb
2
comment Spanning sets and vector spaces
Do you know any other examples of spanning sets? (E.g. for $\mathbb R$ or $\mathbb R^2$.)
Jan
29
comment Linear Transformation from $\mathbb R^n$ to $\mathbb R^m$: image of $1$ dimensional subspace has dimension $1$ or $0$
Can you pinpoint the part of the definition that confuses you? The image of $L$ under $F$ is what is denoted $F(L)$.
Jan
29
reviewed Edit Linear Transformation from $\mathbb R^n$ to $\mathbb R^m$: image of $1$ dimensional subspace has dimension $1$ or $0$
Jan
29
revised Linear Transformation from $\mathbb R^n$ to $\mathbb R^m$: image of $1$ dimensional subspace has dimension $1$ or $0$
added latex
Jan
29
revised Expressing a $SL_2(\mathbb{R})$ matrix as product of…
added 8 characters in body
Jan
29
answered Expressing a $SL_2(\mathbb{R})$ matrix as product of…
Jan
27
comment Fixed point of a map
Right, the question was edited after I wrote the above answer.