Class information -- Math 4153 Finite Dimensional Vector Spaces (Spring 2007)
MWF 10:40-11:30AM, Lockett 237

Syllabus

Homeworks assigned -- Exercises

Section 2.1 -> 3,4,5,9,14,20,26,28,29; due Feb 5, 2007, Mo
Section 2.2 -> 7,12,15,20,21,22,23,24,25,27,29,30,32; due Feb 5, 2007, Mo
Section 2.3 -> 20,21,22,24,27,28,29,32,34,37; due Feb 12, 2007, Mo
Section 2.4 -> 3,5,7,12,13,14,21,23,25,29; due Feb 23, 2007, Fr
Section 3.1 -> 7,8,14,25,29,35,37,39,47,51; due Mar 9, 2007, Fr
Section 3.2 -> 7,9,10,11,15,21,22,23,25,26; due Mar 16, 2007, Fr
I forgot to assign Section 3.3 in class, but will be on the midterm
Section 3.3 -> 8,9,10,11,15,16,19,20,23,25; due Mar 30, Fr 2007, Fr

Miscellaneous Important Notes

Now we have a grader and all the homeworks will be graded.
Midterm 2 is on March 28, Wednesday. Review session is on March 27, Tuesday at 6PM.
Midterm 2 will cover up to p.181 excluding QR factorization.

My class notes

Lectures 15--21 --- Feb28--Mar16 --- pp.37--57
Lectures 22-- --- Mar30--Apr27 --- pp.60--84

Homework grading policy

I do not accept late homework.

Solutions to midterm1

Review material for midterm 1 (ordered wrt most recent)

2.4 -- The four fundamental subspaces

• The four fundamental subspaces.
• Fundamental theorem of linear algebra part I.
• existence and uniqueness of solutions for tall and wide matrices and its relation to the left and right inverses.
• Practice quiz.

2.3 -- Linear independence, basis, and dimension

• (2E) Definition of linear independence
• The columns of A are linearly independent iff N(A) = zero vector
• 2G, 2C
• (2I) A basis for V: (i) The vectors are linearly independent (ii) They span the space.
• Basis and dimension of a vector space. (2K) We proved that the dimension of bases for the same subspace is exactly the same.
• (2L) A basis is a maximal independent set and a minimal spanning set.
• Basis for N(A)= special solutions
  Basis for C(A) = Cols of A (not U) with pivots
  Basis for left nullspace of A = The last m-r rows of L^-1 P where PA = LU
  Basis for row space = Nonzero rows of U (not A)

2.2 -- Solving Ax=0 and Ax=b

• x_complete = x_particular + x_nullspace A x_particular = b; Ax_nullspace = 0
• First step is to check if a solution exists (Consistency condition)
• (2D) If there are r pivots, then there are r pivot variables and n-r free variables. r is the rank of the matrix. m-r rows of U and R are zero then there are m-r consistency conditions.
• rank (AB) <= min(rank(A), rank(B))
• Ax = 0 iff Ux = 0 iff Rx = 0

2.1 -- Vector spaces and subspaces

• A subspace is a vector space which is closed under addition and scalar multiplication.
• Let A be mxn matrix. The column space of A is the subspace containing all the linear combinations of the columns (elements are in R^m). The null space of A is the collection of vectors with Ax=0 (elements are in R^n).

Timeline for class material

(if you missed classes, monitor here what we have done)

Jan 16, 2007, We -- Lecture 01 -- Week 03

2.1 Vector spaces and subspaces