MATH:5010 - Spring 2017

Homework Assignment 4

Reading (from Dummit and Foote):

Read Sections 11.2, 11.3, 11.4, 12.1.

Please look through the sections for EACH lecture before the lecture. See the Lecture Schedule for the sections that will be covered.

Then after EACH lecture, review your lecture notes and thoroughly read the sections covered in class.

Part A Problems (to be handed in on Tuesday, February 14, at the beginning of the lecture in the discussion section):

Section 11.2: # 13, 25. (Note F is a field and all matrices have entries in F.)
IMPORTANT for #25:
- Do NOT use determinants in your proof.
- You may use WITHOUT proof that any matrix can be brought into reduced row echelon form by a finite sequence of elementary row operations (this is basically #16).
- It may be useful to show the following result:
  Let A, U be nXn matrices and let C, W be column vectors with n entries. Suppose the augmented matrix (A|C) is row equivalent to (U|W). Then for all column vectors X with n entries, we have AX = C if and only if UX = W. (It suffices to prove this in the case when A and U differ by an elementary row operation.)
  IF you show this result, PLEASE do NOT say that this follows from the text on p. 424, where it says "The set of solutions in F of this system is not altered..." but make a little argument for each elementary row operation.
Section 11.4: # 4. (Note F is a field and all matrices have entries in F.)
IMPORTANT: You may again use WITHOUT proof that every matrix can be brought into reduced row echelon form by a finite sequence of elementary row operations.

Part B Problems (to be handed in on Friday, February 17, at the beginning of the discussion section in the lecture):

If you hand in ALL Part B Problems together with the Part A Problems already on Tuesday, February 14, 2 extra points will be added to your homework score.

Section 11.3: # 3.
Section 12.1: # 2, 3, 4.
Hint: You may want to do #4 before #3, and then say why #3 follows from #4.

Additional Practice Problems (you have to know how to do these, but you do NOT have to hand these in):

Section 11.2: # 16, 17, 18, 20, 22, 23, 24, 27, 31.
Section 11.4: # 1, 2, 3.
Section 12.1: # 1, 5, 6.

Frauke Bleher
Feb 07 2017