MATH136 Lecture Notes - Linear Combination, Linear Independence, Identity Matrix

• School: University of Waterloo
• Department: Mathematics
• Course: MATH136
• Professor: Robert Sproule

Document Summary

Friday, february 28 lecture 21 : basis and dimension of a vector space. Concepts: define a basis, recognize that any two bases have the same number of elements, verify that a set is linearly independent. Verify that a set is a basis: define dimension of a vector space. 21. 1 definition if {v1, v2, , vk} is both: a spanning family of v and, linearly independent then we say that {v1, v2, ,vk} is a basis of v . Let a = [v1 v2 ,vn], a square n by n matrix. Suppose {v1, v2, , vn} linearly independent . 21. 2 theorem any n linearly independent vectors in the vector space n forms a basis of n. To show that {v1, v2, , vn} is a basis it suffices to show that this set spans n. To show that spans n it suffices to show that col(a) = n.