Linear independence

A set of vectors is linearly independent if only the combination with every coefficient zero sums to the zero vector. Otherwise, the set is linearly dependent.

Linear combination: A sum of vectors multiplied by scalar coefficients. Independence asks whether a linear combination can vanish with any nonzero coefficient.
Vector space: A set with vector addition and scalar multiplication satisfying specified algebraic rules. Linear independence is defined for vectors drawn from a common vector space.
Coordinate system: A rule assigning numerical coordinates to points or vectors relative to chosen reference data. An independent basis gives each vector a unique coordinate representation.
Linear dependence: A property of vectors for which some nonzero coefficient combination equals the zero vector. It is the defining alternative: one dependent subset makes a larger set dependent.
Homogeneous system of linear equations: A system of linear equations whose right-hand sides are all zero. The vectors are independent exactly when the corresponding homogeneous system has only the zero solution.
Scalar (mathematics): An element of a field used to multiply vectors and other mathematical objects. The coefficients in the defining linear combination are scalars from the vector space’s field.
Column space: The subspace spanned by the columns of a matrix. Independent columns form a basis for their column space without redundancy.
Spanning set: A set of vectors whose linear combinations produce every vector in a specified space. Spanning concerns coverage, while independence concerns redundancy; either property can hold without the other.
Rank (linear algebra): The dimension of the column space or row space of a matrix. A matrix’s columns are independent precisely when its rank equals its number of columns.
Zero vector: The additive identity in a vector space, which leaves every vector unchanged when added. The definition tests whether a combination can equal this additive identity.

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