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Coordinate vector spaces
Abstract vector spaces

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Algebraic properties of R^n (or C^n)
Axioms of a vector space
Spans
Linear (in)dependence
Bases
Dimension
Linear transformations
Definition and terminology
Linear combinations
Subspaces
Basic properties
Isomorphism
Geometric properties of R^n (or C^n)
Orthogonality and projection

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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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Linear (in)dependence

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Definition of linearly independent set of vectors: if a linear combination is 0Close match
Definition of linear dependence relation on a set of vectorsClose 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
Definition of linearly dependent set of vectors: one of the vectors can be written as a linear combination of the other vectorsClose match
Theorem: a set of vectors is linearly independent if and only if whenever a linear combination is 0Close 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
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
Two vectors are dependent iff at least one is a scalar multiple of the otherClose match
Linear combinationsClose match
Definition of trivial linear dependence relation on a set of vectorsClose match
A subset of a linearly independent set is linearly independent.Close match
A set of vectors containing more elements than the dimension of the space must be linearly dependentClose match
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.Close match
If a set of vectors contains the 0 vectorClose match
If a set of vectors in R^n (or C^n) contains more than n elementsClose match
Any linearly independent set can be expanded to a basis for the (sub)spaceClose match
Vector spacesClose match
Example of linear combination of vectors in R^2Close match
Linear maps preserve dependence of lists, with image-set caveatsClose match
BasesClose match

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