MATH:5010 - Spring 2017

Homework Assignment 3

Reading (from Dummit and Foote):

Read Sections 11.1, 11.2, 11.3.

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 7, at the beginning of the discussion section):

Section 11.1: # 6, 8, 9. (Note F is a field.)
Hint for #6: Show that for all positive integers i, Ker(φⁱ) ⊆ Ker(φⁱ⁺¹). Since V is finite dimensional, deduce that there is a positive integer m such that Ker(φⁱ) = Ker(φ^m) for all i ≥ m.
Section 11.2: # 15. (Note F is a field.)
IMPORTANT: You may use #14 WITHOUT proof. Read the text on p. 424 after Exercise 13. Also note that two matrices are said to be row equivalent if they lie in the same equivalence class under the equivalence relation introduced in #14.
Hint: It may be helpful to show that the row rank of a matrix A is the same as the dimension of the vector space spanned by the rows of A.
Section 11.3: # 4. (Note that V is an infinite dimensional vector space over a field F.)

Part B Problems (to be handed in on Friday, February 11, at the beginning of the lecture):

If you hand in ALL Part B Problems together with the Part A Problems already on Tuesday, February 7, 2 extra points will be added to your homework score.

Section 11.1: # 10. (Note F is a field.)
IMPORTANT: Note that the null set is the same as the empty set. Follow the hint for the problem. Note that you do not get immediately that a maximal element of S is a basis but you have to make a little argument (use contradiction).
Suppose V is a non-zero vector space over a field F and A is a subset of V that spans V. Prove that A contains a basis of V.
Hint: Do NOT assume A is finite. Modify the proof of Section 11.1, #10, by using a different partially ordered set for Zorn's lemma.
Section 11.2: # 6, 11. (Note F is a field.)

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

Section 11.1: # 1, 2, 4, 5, 7, 11, 12, 13.
Section 11.2: # 1, 2, 3, 4, 5, 7, 8, 9, 10, 14.
Section 11.3: # 1, 3.

Frauke Bleher
Jan 31 2017