FINAL EXAM Study Guide

• See canvas for logistics!

Material Covered

• EVERYTHING IS FAIR GAME!

• Ch 0: review the study guide from Exam 1
• Ch 1 & 2: Euclidean Spaces & Subspaces

• RESTUDY THIS STUFF SINCE I’LL ASK QUESTIONS LIKE THE MATERIAL FROM THIS BUT FOR IPS

• In other words, review the study guide from Exam 1

• Ch 3: Linear Transformations ,

• Again, restudy this since chapter 5 is very similar!
• Definitions from 2.9: can ask about diagonal, triangular, symmetric (), and transposes etc

• Pay close attention to symmetric matrices (iff ). I like using these for “new” problems on the final. No hw is assigned but I’d look at some of the problems from this section. There’s some cool proofs I can ask that are not too long.
• Properties of Transpose: let

• 
• 
• 
•  (notice the order switch!)
• A is invertible iff  is invertible, and

• Ch 4: Vector Spaces & Subspaces

• Definitions, axioms
• Proofs: of basic results, ones with 0 vectors
• Know cardinality, span and basis mean for infinite sets
• Subspaces

• Ch 5: Linear Transformations

• Linear transformations T: V → W

• Matrix representation: [T]_B,B’ = [ [T(v_1) | …. | [T(v_n)] ]
• 1-1, onto, ker, range
• Isomorphisms
• Hopefully, all of this is familiar and it should feel like a “victory lap” you just need to focus on using the new terminology of abstract vector spaces

• Ch 7: Determinants

• Review the Study Guide for Exam 3 on determinants
• Know any method to compute them by hand, especially since it is needed for the eigentheory (ie finding eigenvalues)

• Ch 8: EigenTheory

• Be able to find the eigenvalues by computing determinants
• Be able to find the eigenspaces by setting up a matrix and using RREF (you can use your calculator or ask me to compute it with Sage)
• Be able to find eigenvalues using geometry (i.e. without computations!) for projections and reflections
• EigenTheory and Diagonalization is an important topic. It is extremely likely that I’ll put more problems on the final from this section!

• Ch 9: Inner Product Spaces

• Definitions: inner product (4 axioms), inner product space, norm, unit circle, distance, orthogonal set of vectors, orthonormal set of vectors, orthogonal basis, orthonormal basis, orthogonal complement  of a subspace
• Examples of inner products to know:

• Dot products in R^n and “weighted dot products”
• Using matrices:
• Using integrals: for ,

• Be able to prove classical results:

• Cauchy-Schwarz Inequality, Triangle Inequality (norm version and distance version), Generalize Pythagorean Thm

• Theorems:

• If S is orthogonal (or O.N.) then S is LI. Know how to prove this.

• Be prepared to do Gram-Schmidt Algorithm (see ICA11#2)

• Be able to do this in R^n (n=3, 4, or 5)
• Be able to do this on other IPS, like polynomial space, matrix space, or function spaces.
• Keep in mind: when doing GSA:

• Build orthogonal set first
• Use clever mod along the way (i.e. replace  with a scalar multiple of  to get rid of fractions.
• Build O.N. set at the very end. Ie normalize each  after you’ve built an orthogonal set

• Can ask questions involving  and  in V

• Orthogonal decomposition of a vector :  where  and
• Show
• Dimension Theorem in an inner product space ! Key is that you find an O.N. basis for  and extend it to an O.N. basis for . (statement only)
• Projection mappings:  and

• Know how to find  if you have an O.N. basis for . Use formula:

•  where  is the matrix whose columns are the vectors from the O.N. basis for W only (warning: use only the vectors in a basis for W not the entire basis for V)

• Recall that the eigenvalues and eigenvectors for projection maps are easy to find! They geometry of the maps reveals all!
• Also, projection maps turn out to be diagonalizable.