Linearly independent ↩

Metadata

shorthands: {}
aliases: [Linear independence, linearly independent, linear independence]
created: 2021-11-14 00:12:32
modified: 2022-01-10 04:13:04

is a vector space and

.
The set

is linearly independent if no two distinct finite linear combinations of elements of

can be equal vectors.

Other words: there is only one way to express a given vector using the linear combination of elements in .

A lemma

Statement: If is linearly independent and

Then

Proof

In case of a linearly independent set, having all the coefficients as zeroes is a trivial way to get the zero vector that always works. But due to the definition of linear independence, this is the only way to construct .

Linearly dependent

If is not linearly dependent, then it is said to be linearly dependent.

Properties

If , then is linearly dependent
Proof: in this case we have at least two ways to construct , so cannot be linearly independent anymore.
If is linearly dependent and , then is linearly dependent.
Proof: we have with and not all being zero the same is true for since it has the same elements still.
If is linearly independent and , then is linearly independent.
Proof: this is the contrapositive of the previous.