# Honors Linear Algebra Mathematics 350 — Fall 2015

### Prof. Weibel (640:350:H1)

This course is a proof-based continuation of Math 250, covering Abstract vector spaces and linear transformations, inner product spaces, diagonalization, and canonical forms.
Prerequisites: CALC4, Math 250 and Math 300

Text: Linear Algebra (4th ed.), by Friedberg, Insel and Spence,
Prentice Hall, 2003   ISBN 0-13-008451-4.

• Lectures MW6 (5:00-6:20PM) in ARC 333
• Weibel's Office hours: Monday 3:30-4:45 PM; Wednesday 10:30-12 AM

#### Tentative Course Syllabus

Week Lecture dates  Sections   topics
1 9/2 (W) Chapter 1 Abstract vector spaces & subspaces
29/8 (T), 9 (W)Chapter 1 Span of subsets, linear independence
39/14, 16 Chapter 1 Bases and dimension
49/21, 23 Chapter 2 Linear transformations
59/28, 9/30 Chapter 2 Change of basis, dual spaces
610/5, 10/7 Ch. 1—2  Review and Exam 1
710/12, 10/14 Chapter 3  Rank and Systems of Linear Equations
810/19, 10/21 Chapter 4  Determinants and their properties
910/26, 10/28 Chapter 5  Eigenvalues/eigenvectors
1011/2, 11/4 Chapter 5  Cayley-Hamilton
1111/9, 11/11 Chapter 7  Jordan Canonical Form
1211/16, 11/18 Chapter 7  Rational Canonical Form
1311/23 Ch. 3—5, 7  Review and Exam 2
1411/30, 12/2 Chapter 6  Inner Product spaces
1512/7, 12/9 Chapter 6  Unitary and Orthogonal operators
17 December 21 (Monday) 4-7 PM Final Exam

#### Homework Assignments

HW Due     HW Problems (due Wednesdays
Sept. 161.2 #17; 1.3 #19,23; 1.4 #11,13; 1.5 #9,15
Sept. 231.6 # 20,21,26,29; 1.7 #5,6;
Show that the power series {∑ r^nx^n : r≠0} are linearly independent in F[[x]].
Sept. 302.1 #3,11,28; 2.2 #4; 2.3 #12; 2.4 #15,17
October 72.5 #3(d),7(a,b),13; 2.6 #5,10; Show that F[x]* ≅ F[[x]].
October 213.1 #6,12; 3.2 #5(b,d,h),17; 3.3 #8,10; 3.4 #8,15
If an nxn matrix A has each row sum 0, some Ax=b has no solution.
October 284.1 #10(a,c); 4.2 #23;  4.3 #12,22(c),25(c);  4.4 #6; 4.5 #11,12
Nov. 45.1 #3(b),20,21; 5.2 #4,9(a),12; Show that the cross product
induces an isomorphism between R³ and Λ²(R³).
Nov. 115.2 #18(a),21;  5.3 #2(d,f);   5.4 #6(a),13,19,25
Nov. 187.1 #3(b),9(a),13; 7.2 #3,14,19(a); 7.3 #13,14;
Find all 4x4 Jordan canonical forms of T satisfying T²=T³.
Dec. 76.1; #6,11,12,17;   6.2 #2a,6,11;   6.8 #4(a,c,d),11

Practice problems
Every rank 1 matrix A factors as Fn→F→Fn.
If T is invertible, T and T-1 have the same Jordan form
Find the Jordan canonical form of the 3x3 matrix with all entries 1.
Any T:V→W induces a linear transformation Λ²(V) → Λ²(W).
If dimV=8 and (T+2)i has nullity 3,5,6,7,8 for i=1,...,5 find the Jordan canonical form of T.
If V=F[[x]] and T=d/dx, find the generalized eigenspaces for λ=0 and λ=1.
If T sends R² to itself and has no real eigenvalues, show that (for some basis of R²) T is a rotation followed by a scaling.
If dim(V)=n and i+j=n, show that Λi(V) is the dual space of Λj(V).
Find a bijection between symmetric n-linear forms on Rd and homogeneous symmetric polynomials of degree n in d variables
If Tkv=∑ aiTi for some v, show that g(t)=tk-∑ aiti divides the characteristic polynomial of T.
Show that S={2cos(nt),2sin(nt)}n>0   is an orthonormal set in H and find S.
If v1,...,vn is an orthonormal basis of V, show that <x,y>=∑ <x,vi><y,vi>*
Find the possible Jordan forms for a 3x3 matrix A with A³=A and A²≠I.

Main 350 course page