Math 2406: Abstract Vector Spaces
Lecture 17 Plan
(Tuesday, October 23
Apostol 5.1-5.5.
We begin a new chapter today on determinants, which are scalar-valued functions of square matrices. We'll build towards learning how to compute determinants using the Gram-Schmidt process. A determinant can tell us whether a matrix has an inverse, how many solutions a system of equations may have, and whether a set of vectors is independent. Determinants are also essential for understanding how a given matrix multiplies vectors, as we'll see in the next chapter, Chapter 6, on eigenvalues and eigenvectors.Review questions:
1. How are the three axioms in the definition of
determinant (Sec. 5.3) related to the three
elementary row operations of the Gauss-Jordan process?
2. How can you use the Gauss-Jordan process
to compute the determinant of the 2-by-2 matrix [a b; c d]?
3. Say we have a square matrix where one row is a
scalar multiple of another. What is its determinant and why?