Linearly independent vs. pairwise non-parallel
A list of vectors can avoid redundancy as a whole or one pair at a time. A list of linearly independent vectors has no vector that is a linear combination of the others, which rules out redundancy among three or more vectors as well as among pairs. A list of pairwise linearly independent (pairwise non-parallel) vectors has no two vectors on the same line through the origin.
linearly independent ⊂ pairwise non-parallel
Definitions
A list of vectors is pairwise linearly independent if every two vectors in the list point along different lines through the origin.
Consider a list $v_1, \dots, v_k$ of vectors in a vector space. The list is pairwise linearly independent if, for every two different indices $i$ and $j$, the two vectors $v_i$ and $v_j$ are linearly independent.
Examples
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The vectors $(1, 0)$ and $(0, 1)$ in $\reals^2$
The vectors $(1, 0)$ and $(0, 1)$ are pairwise linearly independent, because neither vector is a scalar multiple of the other.
Non-examples
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The vectors $(1, 0)$ and $(2, 0)$ in $\reals^2$
The vectors $(1, 0)$ and $(2, 0)$ are not pairwise linearly independent, because $(2, 0) = 2 \cdot (1, 0)$.
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The vectors $(1, 0)$ and $(0, 0)$ in $\reals^2$
The vectors $(1, 0)$ and $(0, 0)$ are not pairwise linearly independent, because $(0, 0) = 0 \cdot (1, 0)$ is a scalar multiple of $(1, 0)$.
Properties
- Every linearly independent list is pairwise linearly independent, but a pairwise linearly independent list can be linearly dependent: $(1, 0)$, $(0, 1)$, and $(1, 1)$ are pairwise linearly independent, and $(1, 1) = (1, 0) + (0, 1)$.
- For lists of two vectors, pairwise linear independence and linear independence are the same.
Missing definition: linearly_independent
Examples
| Example | pairwise linearly independent | linearly_independent | Why |
|---|---|---|---|
| The vectors $(1, 0)$ and $(0, 1)$ in $\reals^2$ | ✓ | ✓ | |
Missing example: three_vectors_r2 | |||
| The vectors $(1, 0)$ and $(0, 0)$ in $\reals^2$ | ✗ | ✗ | |
| The vectors $(1, 0)$ and $(2, 0)$ in $\reals^2$ | ✗ | ✗ | |
Notes
- For two vectors, the two notions are the same; the difference appears only for lists of three or more vectors.
- In $\reals^n$, a list of more than $n$ vectors is linearly dependent, but the list can still be pairwise non-parallel, as for every list of distinct directions in the plane.