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How to show the following definitions are identical for an affine space:

  1. $C = p + W$ where $W$ is a subspace $p$ is a vector in $\mathbb{R}^n$, and
  2. $\lambda a + (1-\lambda) b$ is in $C$ for any $a$ and $b$ vectors in $C$ and for any $\lambda$ a real constant

where we are in $\mathbb{R}^n$ as the whole space.

Thank you!

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Suppose that $C=p+W$ and take any $a,b\in C$ and $\lambda\in \mathbb R$. Then $$ \lambda (a-p) + (1-\lambda) (b-p) \in W $$ since $W$ is a vector space. Then $$ \lambda a + (1-\lambda)b \in p + W $$ as desired.

Suppose on the other hand that $$ \lambda a + (1-\lambda) b \in C $$ for all $a,b\in C$ and all $\lambda\in \mathbb R$. Take any point $p\in C$ and define $$ W = C-p. $$ We want to prove that $W$ is a vector space. Take $v,w\in W$ we want to prove that $$ \lambda v + \mu w \in C-p $$ for all $\lambda,\mu \in \mathbb R$. Since $p,p+v,p+w\in C$ also $p+(\lambda+\mu) v = (\lambda+\mu)(p+v) + (1-\lambda-\mu)p \in C$ and $p+(\lambda+\mu) w \in C$. Hence $$ \frac{\lambda}{\lambda+\mu}(p+(\lambda+\mu) v) + \frac{\mu}{\lambda+\mu}(p+(\lambda+\mu w) \in C $$ hence $$ p + v + w \in C $$ i.e. $v+w \in W$.

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