University of Notre Dame
Aerospace and Mechanical Engineering

ME 698: Geometric Nonlinear Control
Homework 2

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

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

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

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

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

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

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

9.
Define the linear operator on $\mathbb R^3$ by

\begin{displaymath}A(x,y,z) = \left( 2 y + z, x - 4 y, 3x \right).
\end{displaymath}

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, 1999.
B. Goodwine (jgoodwin@nd.edu)