What is the difference between linearly and affinely independent vectors?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

The set $\{ v_1,...,v_n\}$ is linearly independent if for general (scalars) $c_1, c_2,...,c_n$: $$\sum_{k=1}^n c_kv_k =0 \Leftrightarrow c_j=0, \: \forall j=1,2,...,n.$$ The set is called affinely linearly independent if they are linearly independent and also $$ \sum_{k=1}^n c_k=1.$$