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I am currently reading a paper titled "Another view of division property" and encountered the terms "Affine subspace" and "Linear subspace". I am new to the field and having some difficulties to understand the topics. Can anyone describeb or give any reference about it so that I can read that.

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In general an affine subspace is not a subspace, it's just a translate (coset) of a subspace. This is because normally we expect $0$ to be in a subspace $V$, since due to closure $x-x \in V.$

If $a+V$ is an affine subspace for $a\neq 0,$ and $V$ a subspace then automatically $a$ is required to be not in $V.$ Otherwise $a+V=V.$

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  • $\begingroup$ One can also visualize this with lines in the space. $\endgroup$ – kelalaka Jun 9 at 7:40
  • $\begingroup$ $a$ could still be in $V$, there is no contraddiction. Indeed ever linear subspace is also an affine one. $\endgroup$ – asd Jun 9 at 9:39
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By "linear subspace" $V$ of some vector space, say $\mathbb{R}^n$, we mean a subgroup of the vector space. So in particular we must have that:

  1. $0\in V$,
  2. For every $v\in V$ we have that also $-v\in V$,
  3. For every $a,b\in \mathbb{R}$ and every $v,w\in V$ we have that $a\cdot v + b\cdot w \in V$.

In other word, in a linear subspace you can do every linear combination of the element of the subspace and you still end up with an element of the same subspace. This is different if we consider "affine subpaces".

An "affine subspace" W of a vector space $\mathbb{R}^n$ can be seen a subset of $R^n$ such that there exists a vector $a$ with the property that $$ W-a=\{v \mid v = w - a \text{ with } w\in W\} $$ is a linear subspace.

From this "definition" we can see that:

  1. every linear subspace is also an affine one (just take $a=0$),
  2. linear combination of elements of an affine subspace need not be in the same affine subspace.
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The other answers provide nice technical definitions. I'm going to provide a more intuitive/visual one.

An affine subspace is a linear subspace plus a translation.

For example, if we're talking about $\mathbb{R}^2$, any line passing through the origin is a linear subspace. Any line is an affine subspace.

In $\mathbb{R}^3$, any line or plane passing through the origin is a linear subspace. Any line or plane is an affine subspace.

Any text or course in Linear Algebra will help you learn more about this if you're interested.

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