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Linear (in)dependence

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Definition of linear dependence relation on a set of vectors
Definition of trivial linear dependence relation on a set of vectors
Determine if a particular set of vectors in R^3 in linearly independent
Definition of linearly independent set of vectors: if a linear combination is 0
Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors
Definition of linearly dependent set of vectors: one of the vectors can be written as a linear combination of the other vectors
Theorem: a set of vectors is linearly independent if and only if whenever a linear combination is 0
A set of vectors is linearly independent if and only if the homogeneous linear system corresponding to the matrix of column vectors has only the trivial solution.
A set of vectors is linearly independent if and only if the matrix of column vectors in reduced row-echelon form has every column as a pivot column.
If a set of vectors contains the 0 vector
Two vectors are dependent iff at least one is a scalar multiple of the other
If a set of vectors in R^n (or C^n) contains more than n elements
A subset of a linearly independent set is linearly independent.
A set is linearly independent if and only if the set of coordinate vectors with respect to any basis is linearly independent.
Removing a linearly dependent vector from a set does not change the span of the set.
Adjoining an element not in the span of a linearly independent set gives another linearly independent set.
Any linearly independent set can be expanded to a basis for the (sub)space

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If a set of vectors contains the 0 vector

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A set is a basis if each vector can be written uniquely as a linear combination.Close match
A set of nonzero vectors contains (as a subset) a basis for its span.Close match
Definition of trivial linear dependence relation on a set of vectorsClose match
A set is linearly independent if and only if the set of coordinate vectors with respect to any basis is linearly independent.Close match
Theorem: a set of vectors is linearly independent if and only if whenever a linear combination is 0Close match
Linear (in)dependenceClose match
An orthogonal set of nonzero vectors is linearly independent.Close match
Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectorsClose match
The 0 vector is unique.Close match
Each vector can be written uniquely as a linear combination of vectors from a given basis.Close match
Definition of the 0/trivial subspaceClose match
A subset of a linearly independent set is linearly independent.Close match
Definition of linearly independent set of vectors: if a linear combination is 0Close match
Any linearly independent set can be expanded to a basis for the (sub)spaceClose match
Definition of linear dependence relation on a set of vectorsClose match
Definition of linearly dependent set of vectors: one of the vectors can be written as a linear combination of the other vectorsClose match
The set containing only 0 is a vector space.Close match
Two vectors are dependent iff at least one is a scalar multiple of the otherClose match
The 0 vector multiplied by any scalar equals the 0 vector.Close match
A set of vectors is linearly independent if and only if the homogeneous linear system corresponding to the matrix of column vectors has only the trivial solution.Close match

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