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

then *g*(x) = K_{1}e^{ x} + K_{2 }e^{ 2x} for some constants K_{1} and K_{2}.

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 L_{v}: V* -> F by L_{v}(T) = T(v) for each T in V*.

i. Show that L_{v} is linear.

ii. Show that L _{α v} = α L_{v} for all α in F and v in V.

(I.e., L_{αv}(T) = α L_{v}(T) for all T in V*.)

iii. Show that L_{v+u} = L_{v} + L_{u} for any v and u in V.

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

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 T_{1} and T_{2} :V → V are linear operators satisfying the
following three properties:

i. For each v in V, T_{1} (v) + T_{2} (v) = v.

ii. For any v in V, T_{1} ( T_{2} (v)) = T_{2} ( T_{1} (v)) = 0 and

iii. For any v in V, T_{1} (T_{1}(v)) = T_{1} (v) and T_{2}(T_{2}(v))= T_{2} (v).

Prove: There are subspace W_{1} and W_{2 }of V such that

V = W_{1} ⊕ W_{2} and for any v in V, T_{1} (v)∈W_{1} and T_{2} (v) ∈ W_{2} .

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)