Linear (in)dependence
Definition: A set of vectors ({\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k}) is linearly dependent if there exist scalars (c_1, \ldots, c_k), not all zero, such that:
[c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k = \mathbf{0}]
The set is linearly independent if the only solution to this equation is (c_1 = c_2 = \cdots = c_k = 0).
Intuition: Linear dependence means at least one vector can be written as a linear combination of the others. Linear independence means no vector is redundant.
Example (dependent): In (\mathbb{R}^2), the vectors ((1,2)), ((2,4)) are dependent because (2(1,2) + (-1)(2,4) = (0,0)).
Example (independent): The standard basis vectors ((1,0)), ((0,1)) in (\mathbb{R}^2) are independent: the only way (c_1(1,0) + c_2(0,1) = (0,0)) is if (c_1 = c_2 = 0).
Key fact: Any set containing the zero vector is linearly dependent. Any set with more vectors than the dimension of the space is linearly dependent.
Parents
Children
- 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