MATH 344                Partnership Problem Assignment                M. FLASHMAN

LINEAR ALGEBRA              50 POINTS                                                 Fall, 2003

** DUE: 5 pm., October 29th.

** UNAUTHORIZED LATE WORK MAY BE PENALIZED.

GROUND RULES:

1. You may consult classmates, notes, textbooks, and myself.

2. You may not consult persons other than those allowed in rule 1.

3. All collaboration and consultation should be acknowledged.

4. Submitted work should reflect your own understanding.

*** You must do all of the following problems.

*** Each problem submitted is worth 10 points.

Note: “Prove” = “Show”

1. Suppose that V is a vector space over ℂ, the complex numbers, with dim V = n.

Prove: V is also a vector space over ℝ, the real numbers, and the dimension of V when considered as a vector space over ℝ is 2n.

2. An excursion into differential equations.

Let V = C (ℝ) = { f :ℝ-->ℝ | f has all order derivatives}, the vector space of real valued C functions defined on the real numbers.

Let D: V → V be the linear transformation defined by (Df)(x) = f '(x) for f in V and x in R. Let T = D - Id and U = D - 2Id .

a. Find the null space of T and the null space of U. (Use calculus for this.)

b. Prove: TU=UT.

c.  Let S = TU. Suppose g is in the null space of S.

Prove: U(g) is in the null space of T and T(g) is in the null space of U.

d. Using parts a. and c., show that if g is a solution to the differential equation

g''(x) - 3g'(x) + 2g(x) = 0,

then g(x) = K1e x + K2 e 2x for some constants K1 and K2.

e. What is the dimension of the null space of S? Explain briefly.

3. Suppose that V is a finite dimensional vector space over the field F. Let V* = L(V,F) and V** = (V*)*.

a. Prove: dim V = dim V* = dim V**.

b. Suppose v ∈V.

Define Lv: V* -> F by  Lv(T) = T(v) for each T in V*.

i.   Show that Lv is linear.

ii.  Show that L α v = α Lv for all α in F and v in V.

(I.e., Lαv(T) = α Lv(T) for all T in V*.)

iii.  Show that Lv+u = Lv + Lu for any v and u in V.

Let E be the function from V to V** defined by E(v) = Lv.

iv. Show that E is linear.

v.   Suppose v is in V and v≠0. Show there is a linear transformation T:V→F where T(v)≠ 0.

vi. Show that E is 1:1 and onto.

vii.         Suppose that L is in V**.

Prove there is a unique vector v in V so that for any T in V*,

L(T) =T(v).

4. Problems from Axler, Linear Algebra Done Right (LA).

a. LA: p 61: 22

b. LA: p 61: 24

5.

a. Suppose that T1 and T2 :V → V are linear operators satisfying the following three properties:

i.   For each v in V, T1 (v) + T2 (v) = v.

ii.  For any v in V, T1 ( T2 (v)) = T2 ( T1 (v)) = 0 and

iii. For any v in V, T1 (T1(v)) = T1 (v) and T2(T2(v))= T2 (v).

Prove: There are subspace W1 and W2 of V such that

V = W1 ⊕ W2 and for any v in V, T1 (v)∈W1 and T2 (v) ∈ W2 .

b. Suppose T is a linear operator on V, T :V → V,

with T(T(v)) = T(v) for all v in V.

Prove:

i.   w ∈ Range(T) if and only if T(w) = w.

ii.  V = Range(T) ⊕ Null(T)