### ME 698: Geometric Nonlinear Control Homework 2

 B. Goodwine Fall, 1999 Issued: September 2, 1999 Due: September 9, 1999

1.
Let be the set of linear transformations between the vector spaces V and W. Show that if then
(a)
the range of L, R(L) is a subspace of W, and
(b)
the null space of L, is a subspace of V.

2.
Let U1 and U2 be subspaces of the vector space V.
(a)
Show that is a subspace of V.
(b)
Show by counterexample that need not be a subspace.

3.
Let

Are u and v linearly independent?

4.
Let V be a vector space and let be a linearly independent set. Indicate whether each of the following are linearly dependent or linearly independent:
(a)
(b)
(c)
.

5.
Let V be the vector space C[0,T]. Which of the following subsets of V are subspaces?
(a)
(b)
(c)
(d)
(e)

6.
Let U,V and W be vector spaces over and let and be linear. Show that the operator is linear.

7.
Let V be a vector space. A set is convex if

Let be two convex sets.

(a)
Show is convex.
(b)
Is convex?

8.
Let V be a vector space and (the set of linear operators on V. Show that

9.
Define the linear operator on by

Find the matrix representation of A in the bases
(a)
, and
(b)
.

I would rather discover a single fact, even a small one, than debate the great issues at length without discovering anything at all.

--- Galileo Galilei.

Last updated: September 2, 1999.
B. Goodwine (jgoodwin@nd.edu)