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Questions tagged [quotient-spaces]

Quotient space is a space where the underlying space is set of equivalence classes of some equivalence relation. As this concept appears in various areas, include also a tag specifying subject matter, such as (topology), (vector-spaces), (normed-spaces), etc.

Quotient space is a space where the underlying space is set of equivalence classes of some equivalence relation. This concept appears in various contexts. For example, quotient spaces can be defined for topological spaces, vector spaces and normed spaces.

As this concept appears in various areas, include also a tag specifying subject matter, such as , , , etc.

1916 questions
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Why does the quotient space V/W not equal the vectorspace V?

Let's say $V=\Re^3$ and $W$ is a plane through the origin. The way I understand the quotient space $V/W$ is that it's formed by taking every vector $\vec{v}^{\,} \in V$ and adding it to the subspace $W$. Why doesn't this make $V/W = V$?
Joshua
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Induced Bilinear Form on $V/A$

We are given a matrix \begin{bmatrix} 2&3&3&0\\ 3&4&3&-1\\ 3&3&0&-3\\ 0&-1&-3&-2 \end{bmatrix} and we want to find the induced bilinear form on $V/A$ where $A = span\{3e_1-2e_2+e_4, 3e_1-3e_2+e_3\}$ I'm not really sure where to start with this…
twister
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CRC calculate quotient

As above, i have absolutely no idea how to calculate the quotient (10110110). Some mentioned that there is no need for it, but my exams required me to understand how to get the quotient. Please help thanks.
user3807187
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Existence of an element from a quotient space

Let $V$ be a finite-dimensional complex inner product space and $U\subset V$ , a subspace of $V$ and $v \in V $, then it exists a unambiguous $w \in v + U$, such that $$\Vert w\Vert= min\{\Vert v'\Vert : v' \in v+U\}$$ While I know that this $w$ is…
user1072059
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How to show $n$-torus is compact

Set $\Bbb T^n = \Bbb R^n \setminus \Bbb Z^n$. Define $d : \Bbb T^n \times \Bbb T^n \to \Bbb R$ by $$d(x+\Bbb Z^n,y+\Bbb Z^n)=\inf\{\|v-w\|:v\in x+\Bbb Z^n, \text{ and } w \in y+ \Bbb Z^n\}$$ Show that $ \Bbb T^n$ is complete and compact My thought :…
fivestar
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Algorithm to find a basis of a quotient space $R^n/R^m$.

I have a set of $m$ vectors $\{x_i\}$, $x_i \in R^n$. How can I obtain a basis for $R^n/span(\{x_i\})$?
Him
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Why do topological quotients "bend" lines?

Why do topological quotients "bend" lines? http://mathonline.wikidot.com/topological-quotients-in-euclidean-space I have no problem with the idea that one constructs a topology on the line from its subsets or that the equivalence relation $0 \sim 1$…
mavavilj
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