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 R^{2}is the unit sphere of a

unique inner product.

2. Let || || be a norm on R^{2}. Show that there is an inner product ( , )

on R^{2}satisfying

|| v || ≤ (v,v)^{1/2}≤ K || v ||

for some constant K ≥ 1 for all v in R^{2}.

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 v_{1}, . . . , v_{n}be a basis for V.

Let A: V ---> W be an isomorphism.

Show that Av_{1}, . . . , 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}, R^{n})? 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 : R^{n}---> R^{m}be a linear transformation.

Show that if n > m, then T is not left invertible.

3. Again let T : R^{n}---> 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.