(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

So I'm a little confused about what a tangent space is. Is it the same as the equation of the tangent plane in lower dimensions?

My notes define the tangent space as follows.

Let M be a hypersurface of R^{d}.

Let x(s) be a differentiable curve in M such that x(0)=x_{0}is in M.

Then x'(0) is a tangent vector.

Thetangent spaceat x_{0}of M is

T_{xo}M= { v in R^{d}: there exists a curve x(s): [-1,1]->M such that x(0)=x_{0}, x'(0)=v}

T_{xo}M is a vector space of the same dimension as M.

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Now I don't understand why T_{xo}M is a vector space. A vector space MUST contain the zero vector, which is the origin.

But even in lower dimensions, say M is the unit sphere in R^{3}, the tangent plane at (0,0,1) clearly does not pass through the origin (0,0,0). How can this be a vector space?

2. Relevant equations

Tangent Space

3. The attempt at a solution

N/A

Any help/clarifications would be much appreciated!

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# Tangent space and tangent plane

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