1. [IS%] Consider the bases B = {(1,2,4), (-1,2, O), (2,4,0)} and
B' = { (0,2,1$, (-2,1,O), (I, I, I)} of R ~ . Find the transition matrix from 5 to 5'.
2. [lo%] Find an orthonormal basis of the subspace of R~ spanned by the vectors (1.~1,-I) and (X,O,-I).
3. [lo%]
(a) (5%) True or False: If A is diagonalizable, then the rank of A equals the number of nonzero eigenvalues of A.
(b) (5%) Prove your answer of (a).
4. [15%]
(a) (1 0%) P2 is a set including all polynomials of degree less than 2. Let A
c
P2 be the vector space spanned by polynomials x2+
2x+
1, x + 2, x2 - x - 1 and let Bc
P2 be the vector space spanned by 1, x, x2. Describe the difference setA-B =(f(x): f (x)EA, f ( x ) ~ B } .
Note: The space spanned by polynomials includes all polynomials that can be written as a linear combination of those polynomials.
(b) (5%) Letfi(x) E x2 + 2x
+
1. Consider the "multiplication" defined by the logical AND, and "addition" defined by the logical OR. That is, we definex = X . X 2 = x ~ x ( x A N D x ) , 2 x = x + x = x v x ( x O R x ) , a n d x + c = x v c . Evaluate3 (FALSE=O) andfi(TRUE=l).
(a) (7%) Compute eigenvalues and the corresponding eigenvectors of A.
(b) (8%) Compute lim A'".
n+m
6. [6%] How many ways are there to pack eight identical DVDs into five indistinguishable boxes so that each box contains at least one DVD? Explain why the answer you have.
7. [6%] Determine whether the function f (x) = (x
+
1) l ( x+
2) is a bijection from R to R.Explain why the answer you have.
8. [12%] Suppose that the function f satisfies the recurrence relation
f (n) = 2 f (&)
+
log, n whenever n is a perfect square greater than 1 and f (2) = 1.(a) (4%) Find f (16).
(b) (8%) Find a big-0 estimate for f (n).
9. [11%]
(a) (4%) Find the state table for the nondeterministic finite-state automaton with the state diagram as shown in Figure 1.
(b) (7%) Find a deterministic finite-state automaton that recognizes the same language as the nondeterministic finite-state automaton in (a).
Start -
Figure 1
