Summary: Linear Algebra Worksheet 7
Math 108A Fall 2009, TA Grace Kennedy
Course Website: http://math.ucsb.edu/kgracekennedy/F09108A.html
Supplemental Reading: Axler Chapter 5
Invariant Subspaces, Eigenvalues and Eigenvectors
Let F be a field, V be a vector space of dimension n < over F, and let L(V )
be the set of linear transformations from V to V . Unless otherwise stated, all
vector spaces are over this field F. Also, let T, S L(V )
1. What is the dimension of L(V )?
2. Prove or disprove that the following are invariant subspaces:
(a) The range of T under T.
(b) The range of S under TS.
(c) The span of any set of eigenvectors of T under T.
(d) The span of any set of eigenvectors of T under TS.
3. How many eigenvalues can the linear transformation from Midterm 2 have?