A geographic example may help to clarify the concept of linear independence.

The concepts of linear independence, span, basis, and dimension (which is called rank in module theory) still make sense.

In his work, the concepts of linear independence and dimension, as well as scalar products are present.

For linear equations, logical independence is the same as linear independence.

Allowing more linear combinations in this case can also lead to a different concept of span, linear independence, and basis.

This was a result which established the algebraic independence of classes of numbers rather than just linear independence.

Hence we have linear independence.

Tutorial and interactive program on Linear Independence.

Since we have a repeated root, we have to introduce a factor of x for one solution to ensure linear independence.

On the contrary, linear independence is a mere instance of present independence, which becomes equivalent to it in such vector spaces.

By the Gram-Schmidt process, it is the same to ask for (pointwise) linear independence or fields that give an orthonormal basis at each point.

Online Notes on Linear Independence.

(Also, several other generalizations of linear independence for universal algebras do not imply present independence.)

When the replace the , both the linear independence and spanning properties for infinite basis sets follow from present outer condition and conversely.

Introduction to Linear Independence at KhanAcademy.

So, to get linear independence all non zero -tuples with one as a top most non zero element will be chosen as columns.

Baker's theorem grants us the linear independence over the algebraic numbers of logarithms of algebraic numbers.

Matrix methods are significantly limited for MIMO systems where linear independence cannot be assured in the relationship between inputs and outputs.

(Since the manifold metric is always assumed to be regular, the compatibility condition implies linear independence of the partial derivative tangent vectors.)

At least one of must be non-zero, otherwise this equality would contradict the linear independence of ; note that this additionally implies that .

The real numbers form a vector space over the rational numbers, and this is equivalent to the usual definition of linear independence in this vector space.

Clearly, any block-diagonal matrices constructed from involutory matrices will also be involutory, as a consequence of the linear independence of the blocks.

Furthermore, linear independence is only defined as the property of a collection; it therefore is important if those vectors are linearly independent as a set or as a family.

In the mathematical theory of matroids, a matroid representation is a family of vectors whose linear independence relation is the same as that of a given matroid.

Linear independence constraint qualification (LICQ): the gradients of the active inequality constraints and the gradients of the equality constraints are linearly independent at .