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
- A set of two vectors is linearly dependent if and only if neither 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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