Math 2406: Abstract Vector Spaces
Lecture 13 Plan
(Thursday, October 4
Guest lecturer: Professor
Guillermo Goldsztein
Apostol 4.7-4.9.
We have seen that one-to-one maps have nice properties: they are invertible and have a trivial null space. We'll use these two properties to show they are uniquely characterized by how they transform bases. What's more, we can use this property to understand a matrix more fundamentally: it transforms coefficients of elements in a basis of the domain into coefficients of elements in a basis of the range.Review questions:
1. How does the fact that a
one-one map has a trivial null space enter into the proofs of Theorems
4.11 to 4.13?
2. We can specify a linear transformation uniquely based on how it
transforms a basis into another set (not necessarily a basis).
When we introduce matrices in Theorem 4.13, we only talk about
transformations from one basis to another basis. What if {w} is not a
basis--does the Theorem still hold? Think of an example in R^2.
3. How do we know c implies b in Theorem 4.12?
No
homework is due October 11.