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What is the difference between linearly and affinely independent vectors? Why does affine independence not imply linear independence necessarily? Can someone explain using an example?

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To augment Lord Shark's answer, I just wanted to talk a little about the intuition behind it.

Intuitively, a set of vectors is linearly dependent if there are more vectors than necessary to generate their span, i.e. the smallest subspace containing them.

On the other hand, a set of vectors is affinely dependent if there are more vectors than necessary to generate their affine hull, i.e. the smallest flat (translate of a linear space) containing them.

A single vector $v$ in a vector space generates an affine hull of $\lbrace v \rbrace$, which is just the trivial subspace $\lbrace 0 \rbrace$ translated by $v$. But, if $v \neq 0$, the span is the entire line between $0$ and $v$, as $0$ must be part of any subspace. To generate that line as an affine hull, you could look at the list $v, 0$.

So, $v, 0$ are linearly dependent (e.g. $0 = 0 \cdot v + 5 \cdot 0$) as $0$ is not necessary to generate the span (just $v$ would have done fine), but both are necessary to generate the line as the affine hull, so they are affinely independent. To prove this, suppose $\lambda_1 + \lambda_2 = 0$ and,

$$\lambda_1 \cdot v + \lambda_2 \cdot 0 = 0.$$

Then $\lambda_1 \cdot v = 0$, which implies $\lambda_1 = 0$, since $v \neq 0$. Since $\lambda_1 + \lambda_2 = 0$, we therefore also have $\lambda_2 = 0$. This proves affine independence.

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Vectors $v_1,\ldots,v_n$ are linearly dependent if there are $\lambda_1,\ldots,\lambda_n$, not all zero, with $\lambda_1 v_1 +\cdots+\lambda_n v_n=0$.

Vectors $v_1,\ldots,v_n$ are affinely dependent if there are $\lambda_1,\ldots,\lambda_n$, not all zero, with $\lambda_1 v_1 +\cdots+\lambda_n v_n=0$ and $\lambda_1+\cdots+\lambda_n=0$.

In $\Bbb R^1$ any two distinct vectors are linearly dependent but affinely independent.

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The set $\{ v_1,...,v_n\}$ is linearly independent if for $c_1, c_2,...,c_n$, not all zero, such that $$\sum_{k=1}^n c_kv_k =0 \Leftrightarrow c_j=0, \: \forall j=1,2,...,n.$$ The set is called affinely linearly independent if they are linearly independent and also $$ \sum_{k=1}^n c_k=1.$$

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