MATH 4377 Lecture Notes - Lecture 3: Triangular Matrix, The Algorithm, Linear Combination
Document Summary
Week 3 math 4377 and math 6308. Section 1. 5: linearly dependent sets and linearly independent. Note: from now on, unless otherwise stated, saying. (cid:1848) is a vector space is the same as saying dependent if and only if there exists a finite number of vectors. (cid:1848) is a vector space over a field (cid:1832) . A subset (cid:1845) of (cid:1848) is linearly (cid:1873)(cid:2869), ,(cid:1873)(cid:3040) in s and scalars (cid:1855)(cid:2869), ,(cid:1855)(cid:3040), not all zero, so that (cid:1855)(cid:2869)(cid:1873)(cid:2869)(cid:3397) (cid:3397)(cid:1855)(cid:3040)(cid:1873)(cid:3040)(cid:3404)(cid:882) Also, we say the vectors (cid:1873)(cid:2869), ,(cid:1873)(cid:3040)(cid:1488)(cid:1848) are linearly dependent if and only if the set (cid:4668)(cid:1873)(cid:2869), ,(cid:1873)(cid:3040)(cid:4669) is a linearly dependent set. Definition: a subset s of a vector space v is said to be linearly independent if and only if it is not linearly dependent. Remark: by default, the empty set is a linearly independent subset of any vector space (cid:1848). Note: some of the examples below contain results that you might want to use.
