University of Notre Dame
Aerospace and Mechanical Engineering

ME 654: Geometric Nonlinear Control
Homework 2

B. Goodwine Fall, 2003

Issued: August 28, 2003 Due: September 11, 2003

1.

Let $\mathcal{L}(V,W)$ be the set of linear transformations between the vector spaces V and W. Show that if $L \in \mathcal L(V,W)$ then

(a)

the range of L, R(L) is a subspace of W, and

(b)

the null space of L, $\mathcal N (L)$ is a subspace of V.

2.

Let U₁ and U₂ be subspaces of the vector space V.

(a)

Show that $U_1 \cap U_2$ is a subspace of V.

(b)

Show by counterexample that $U_1 \cup U_2$ need not be a subspace.

3.

Let

$$u = \left[ \begin{array}{c} 1 \\ -3 \\ 2 \end{array} \right] ... ... v = \left[ \begin{array}{c} 2 \\ -1 \\ 2 \end{array} \right].$$

Are u and v linearly independent?

4.

Let V be a vector space and let $\{ x_1, x_2, x_3 \}$ be a linearly independent set. Indicate whether each of the following are linearly dependent or linearly independent:

(a)

$\{x_1, x_3 \}$

(b)

$\{ x_1, x_1 + x_2, x_1 + x_2 + x_3 \}$

(c)

$\{0, x_1, x_2, x_3\}$.

5.

Let V be the vector space C[0,T]. Which of the following subsets of V are subspaces?

(a)

$B_1 = \{ f \in C[0,T] \quad \vert \quad f(0) = f(T) \}$

(b)

$B_2 = \{ f \in C[0,T] \quad \vert \quad f(0) = f(T) = 0 \}$

(c)

$B_3 = \{ f \in C[0,T] \quad \vert \quad f(t_1) = f(t_2) \forall t_1,t_2 \quad \mbox{such that} \quad t_1 + t_2 = T \}$

(d)

$B_4 = \{ f \in C[0,T] \quad \vert \quad f(0) = 1 \}$

(e)

$B_5 = \{ f \in C[0,T] \quad \vert \quad \int_0^T f(\tau) d \tau = 1 \}$

6.

Let U,V and W be vector spaces over $\mathbb R$ and let $L_1 \in \mathcal L (U,V)$ and $L_2 \in \mathcal L (V,W)$ be linear. Show that the operator $L_2 L_1 \in \mathcal L (U,W)$ is linear.

7.

Let V be a vector space. A set $K \subset V$ is convex if

$$\lambda x + (1 - \lambda) y \in K \qquad \mbox{for} \qquad \lambda \in [0,1], \quad x,y \in K.$$

Let $K_1, K_2 \subset V$ be two convex sets.

(a)

Show $K_1 \cap K_2$ is convex.

(b)

Is $K_1 \cup K_2$ convex?

8.

Let V be a vector space and $A \in \mathcal L (V)$ (the set of linear operators on V. Show that

$$A \quad \mbox{is invertible} \quad\Longleftrightarrow \quad \mathcal N (A) = \{ 0 \}.$$

9.

Define the linear operator on $\mathbb R^3$ by

$$A(x,y,z) = \left( 2 y + z, x - 4 y, 3x \right).$$

Find the matrix representation of A in the bases

(a)

$B_1 = \{ (1,1,1),(1,1,0),(1,0,0) \}$, and

(b)

$B_2 = \{ (1,0,0),(0,1,0),(0,0,1) \}$.

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, 2003.
B. Goodwine (jgoodwin@nd.edu)