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The book "Introduction to Linear Algebra" by Gilbert Strang says that every time we see a space of vectors, the zero vector will be included in it.

I reckon that this is only the case if the plane passes through the origin. Else wise, how can a space contain a zero vector if it does not pass through the origin?

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  • $\begingroup$ If it's a vector space (over a field), the all-zero vector has to be included, because you can multiply any vector by the zero element of the field and the result has to be in the vector space $\endgroup$ – Luis Mendo Jul 28 '17 at 10:20
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    $\begingroup$ When he says "space of vectors" he most probably meant "vector space" which is by definition always passing through zero (in some sense). Otherwise its called an "affine space". $\endgroup$ – M. Winter Jul 28 '17 at 10:21
  • $\begingroup$ Related: Zero vector of a vector space. $\endgroup$ – Andrew D. Hwang Jul 28 '17 at 12:16
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This is part of the definition of "vector space", which requires addition and so must have a zero vector for addition to be useful.

There are other things that can be called a "space" though. What you're thinking of is called an affine space: https://en.wikipedia.org/wiki/Affine_space

In particular, affine spaces don't have any way of adding vectors. Consider the plane in 3D given by $x=1$. If I try to add the points $(1,5,9)$ and $(1,-3,0)$, I get $(2,2,9)$, which is no longer in the plane. The fact that I can't add points is what makes it not a vector space.

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A plane not passing through the origin is not a vector (sub)space: it has no $0$ vector, nor the sum of two vectors in the plane belongs to the plane.

However such a plane has another structure, known as an affine (sub)space. Roughly, it means that if you choose a vector $u_0$ in the plane $\Pi$, the set $$\Pi_0=\{u-u_0\mid u\in \Pi\}$$ is a vector subspace.

In other words, an affine plane is a translated $u_0+\Pi_0$ of a vector subspace $\Pi_0$, which is unique (it does not depend on the choice of $u_0$). $\Pi_0$ is called the direction of the affine plane $\Pi$.

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