MATH 21 Lecture Notes - Lecture 5: Coefficient Matrix, Linear Combination, Subsequence
Document Summary
Dependence relation: sequence of scalars such that linear combinations c1v1 + c2v2 + . + ckvk is equal to the zero vector. Trivial dependence relation: the particular dependence relation (0, 0, . Linearly dependent: if there exists a non-trivial dependence relation for a sequence of n-vectors. Linearly independent: if only dependent relation for the sequence of vectors is the trivial dependence relation. Given 3 vectors, create the coefficient matrix to determine which columns are pivot columns. Then find the red of the matrix to find the general solution of the homogeneous linear system (hls) The general solution helps determine if the vectors are linearly independent or dependent. Theorem: let (v1 . vk) be a sequence of n-vectors. The following are equivalent: (v1, v2 . vk) are linearly independent, the hls coefficient matrix has only the trivial solution, the matrix a has k pivot columns, that is, each column of a is a pivot column.
