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Definition: Let $v_0, v_1.. v_k$ be points in $\mathbb{R}^d$. These points are called affinely independent if there do not exist real numbers $\alpha_0, \alpha_1...\alpha_k$ that are not all zero such that $\sum_{i=0}^k \alpha_i v_i = 0$ and $\sum_{i=0}^k \alpha_i = 0$.

We need to prove the following: The points $v_0, v_1... v_k$ are affinely independent if and only if the vectors $v_1 - v_0, v_2 -v_0... v_k - v_0$ are linearly independent.

Thank you so much.

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You can try expressing one of the coefficients, say the first one, with the remaining ones since $\sum_{i=1}^{k-1} \alpha_i = -\alpha_0$. Then plug this into the definition of affine dependence and you will get a pattern. – user13838 Nov 15 '11 at 0:52

Assume affine independence. Assume some linear combination of the $v_i-v_0$ is 0. Write that linear combination down (with unknown coefficients) and manipulate it to a form where you can use the assumption of affine independence.

Conversely, assume linear independence. Let some linear combination of the $v_i$ be zero. Do some manipulation to relate it to a linear combination of the $v_i-v_0$. Use the linear independence hypothesis to draw a conclusion about the coefficients.

If all that is too abstract for you, try to do the case $k=1$ or $k=2$ where you can more easily see what's happening.

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