Aerospace and Mechanical Engineering

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
*U*_{1}and*U*_{2}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.

B. Goodwine (