Honors Linear Algebra (Math 0540 S02)

with Richard Kent.

Office: KH 313

Office hours: Tuedays 3–4pm, Wednesdays 11am–noon.



Text: Linear Algebra Done Wrong by Sergei Treil.



Handout.


Final exam: 2:00 PM on May 13, 2008 in Barus & Holley 163.


Notice: We will observe Reading Period until May 6, when we will
meet at the usual time and place to review for the final.

Bring questions!



Homework:


HW 9, due Thurs. April 24:
From Chapter 5:
To turn in: 1.1–1.9.
Not to turn in: 2.1–2.9, 3.1–3.8, and the following:

1. Prove that an ellipse centered at 0 in R² is the unit sphere of a
unique inner product.

2. Let ||   || be a norm on R². Show that there is an inner product (  ,  )
on R² satisfying
                                || v || ≤ (v,v)^1/2 ≤ K || v ||

for some constant K ≥ 1 for all v in R².




HW 8, due Thurs. April 10:
From Chapter 4: 1.1–1.11, and 2.1–2.14.




HW 7, due Thurs. April 3:
Read the rest of Chapter 3, from Chapter 3: 5.1–5.6, and 7.1–7.5.




HW 6, due Thurs. March 20:
From Chapter 3: 3.1–3.12, and 4.1–4.4.




HW 5, due Thurs. March 13:
From Chapter 2: 6.1, 6.2, 7.1–7.9.




HW 4, due Thurs. Feb. 28:
From Chapter 2: 2.1, 3.1–3.8.

You should do the following, but you don't have to turn them in:
5.1–5.5.




HW 3, due Thurs. Feb. 21:
3.3c), 3.3d), 5.1–5.12, 6.1, and the following:

1. Let A: V ---> W be an isomorphism. Show that A is surjective.

2. Let V be a vector space and let v₁, . . . , v_n be a basis for V.
Let A: V ---> W be an isomorphism.
Show that Av₁, . . . , Av_n is a basis for W.

3. Let A: V ---> W be a linear transformation.
Suppose that there is a function B: W ---> V such that A(B(w)) = w for all w in W
and B(A(v)) = v for all v in V (so B is a set theoretic left and right inverse for A).
Show that B is a linear transformation.

4. What's the dimension of Hom(R^m, Rⁿ)? Find a basis.




HW 2, due Tues. Feb. 12:
3.1, 3.2, 3.3a), 3.3b), 3.4, 3.5, 4.1–4.7, and the following:

1. Let T : V ---> W be a linear transformation.
Prove that if ker(T) contains a nonzero vector, then T is not left invertible.

2. Let T : Rⁿ ---> R^m be a linear transformation.
Show that if n > m, then T is not left invertible.

3. Again let T : Rⁿ ---> R^m be a linear transformation.
Show that if n < m, then T is not right invertible.

4. Find matrices A and B that are not invertible such that AB is invertible.

5. Let A and B be matrices whose product AB is defined.
Show that if two of A, B, and AB are invertible, then the other one is.




HW 1, due Tues. Feb. 5:
1.2–1.4, 2.2, 2.4–2.6, and the following:

Let V and W be vector spaces.

1. Show that the intersection of two subspaces of V is also a subspace.

Let T:V ---> W be a linear transformation.

2. Let U be a subspace of V. Show that T(U) is a subspace of W.

3. Let U be subspace of W. Show that T^–1(U) is a subspace of V.

4. Show that Hom(V,R) is a vector space.