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| 1 | Scalar |
| 1 | Vector Space |
| 1 | Vector |
| 1 | Field |
| 1 | Finite |
| 1 | Set |
| 1 | Set Difference |
| 1 | Infinite |
| 1 | Ring |
| 1 | Module |
| 2 | Characterization |
| 3 | Linear Combination |
| 5 | Eigenvector |
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Linear Independence
Let
be a vector space
over a
field
. We say that
are linearly dependent if there exist scalars
, not all zero, such that
In the case of two vectors, linear dependence means that one of the vectors is a scalar multiple of the other. As an alternate characterization of dependence, we also have the following.
Remark. Linear independence can be defined more generally for modules
over rings: if
is a (left) module over a ring
. A subset
of
is linearly independent if whenever
for
and
, then
.