Homework 8
Math 121A, Lecture A, S’08
In what follows, U, V, W are vector spaces over the same field F. Specific choices
of the spaces and of Fare made in some of the problems.
1. A linear transformation A∈ L(U, V ) is said to be of finite rank if the range
Ran(A) of Ais finite-dimensional. If Ais of finite rank, then the dimension of
Ran(A) is called the rank of Aand (in this course) is denoted Rk(A). If the
kernel of Ais finite-dimensional, the dimension is called the nullity of Aand
(in this course) denoted Nlty(A).
Prove that if Uis finite-dimensional, then dim(U) = Rk(A) + Nlty(A).
2. Suppose Vis finite-dimensional, X={~x1. . . , ~xn}is a basis of V, and X0=
{~x0
1. . . , ~x0
n}is the dual basis. The matrix of a linear transformation A∈ L(V)
in the basis Xis known to be [A]X= (αi,j)i,j . Find the matrix of A0in X0.
How are det(A) and det(A0) related?
3. Recall that, if Sis a subspace of V, the annihilator of Sis the set of all
functionals f∈V0such that
[~s, f ] = 0 for all ~s ∈S
The annihilator of Sis denoted S0.
Let V=U⊕W, and suppose A∈ L(V) is a projection on Ualong W. Prove
that A0is the projection on W0along U0.
THE REST OF THIS HOMEWORK ASSIGNMENT IS OPTIONAL.
ITS SOLE PURPOSE IS TO EXPOSE THOSE INTERESTED TO
WEAK FORMULATIONS (OCCURRING IN FUNCTIONAL ANAL-
YSIS, FINITE ELEMENTS), LINEAR ORDINARY DIFFEREN-
TIAL EQUATIONS, AND CONTROL THEORY.
4. Let ~y be a fixed element of V, and let A∈ L(V). One often seeks at least one solution ~x of
the equation
A~x =~y (1)
(a) Prove that ~x is a solution of (1) if and only if
[~x, f ]=[~y , A0f] for all f∈V0,(2)
where A0is the adjoint of A.
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