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 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 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)