Question 1: What are the 1-, 2-, and infinite norm of vectors
x1
=
−3
, x2
=
1
2
1
−1
−1
Question 2: Find two orthonormal vectors that span the same space as two vectors in problem 1.
Question 3: Find the rank and nullities of the following matrices
A1
=
0
0
0
, A2
=
3
2
0
, A3
=
0
−1 2
2
0
1
0
4
1
1
1
2
−3
4
0
0
−1
1
1
0
0
0
0
1
Question 4: If A matrix is given as follows
1 1 0
A1= 0 0 1
0 0 1
Then compute A10, A103, eAT
Question 5: Find the unit step response of the following system using two different methods
˙
0 1
X(t) +
1
u(t)
X(t) =
−2 −2
1
y(t) = [2 3]X(t)
Question 6: Are the two sets of state-space equations
X˙(t) =
0 2
2
X(t) +
1
u(t), y(t) = [1 − 1 0]X(t)
2
1
2
1
0
0
1
0
X˙(t) =
0 2
1
X(t) +
1
u(t), y(t) = [1 − 1 0]X(t)
2
1
1
1
0
0
−1
0
equivalent? Zero state equivalent?
