注意：務必請按照順序答題，以避免發生誤改或誤判。
1.	(10%) What linear combination of can form ?
2.	(20%) Find the standard matrix A for a linear transformation on ; the transformation is a reflection about the xy-plane, followed by a clockwise rotation of 30° about the positive z-axis. Then, determine the inverse matrix of A .
3.	(20%) Determine the eigen values and spaces of the following matrix B , and then determine .
4.	Here are the rules of our national lottery.
	＊For each ticket that you purchase, you must pick 6 distinct numbers from 1 through 42.
	＊The cost of each ticket is 50 NT.
	＊The host will randomly select 6 distinct numbers from 1 through 42.
	＊If the six numbers that you select are the same as the six numbers that the host selects, then you win the first prize.
	Suppose Mr. Chen has selected 10 distinct numbers from 1 through 42 (e.g., 1, 12, 23, 34, 5, 16, 27, 38, 9, 10). He wants to buy all lottery tickets with these numbers, and no two tickets should have the same set of numbers. For example, a ticket with the following numbers will be what Chen wants: 1, 12, 5, 38, 9, 10.
	(a) (5%) How much will Chen have to pay in order to get all these tickets?
	(b) (5%) What is Mr. Chen's chance of winning the first prize?
5.	(5%) What is the adjacency matrix for the following graph?
	The set of nodes : {a, b, c, d, e}
	The set of edges : { (b, d), (b, c), (a, e), (d, a), (c, e) }
6.	(5%) Please complete the following definition of a relation. Let A be a non-empty set. Let B be a non-empty set. A relation R between A and B is such that ______________________.
7.	(5%) = is a relation. In fact, it is a reflexive relation. Because 2=2, 3=3, 45=45, etc.
	What is the definition of a reflexive relation R?
8.	(5%) Give a real life example (i.e., give the name of the "thing") that is the realization of a finite state machine.
9.	(a) (5%) What is the halting problem?
	(b) (5%) The halting problem is well known for its unsolvability.
	What does 'unsolvable' mean? Give a complete definition of it.
10.	(10%) Please state the differences between the following three "things" : tree, forest, directed acyclic graph.

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