MATH125 Lecture Notes - Lecture 21: Invertible Matrix, Identity Matrix, Scalar Multiplication

• School: University of Alberta
• Department: Mathematics
• Course: MATH125
• Professor: Nikita

Document Summary

This section provides a review of di erent criteria when a square matrix a is invertible. The invertible matrix theorem divides the set of all n n matrices into two disjoint classes: the invertible (or nonsingular) matrices and the non invertible (or singular) matrices. This section focuses on perhaps most important ideas in the entire course: the notions of a subspace, a basis, rank and dimension. The no- tion of a subspace is simply an algebraic generalization of the geometric examples of lines and planes through the origin. The fundamental con- cept of a basis for a subspace is then derived from the idea of direction vectors for such lines and planes. In words, we say that h is closed under addition and scalar multiplication and contains zero. Then span { u, v } is a subspace in rn. In- deed, we need to verify three properties.