臺北巿立師範學院 九十三學年度研究所碩士班入學考試試題

所　　別：數學資訊教育研究所

科　　目：基礎數學

考試時間：九十分鐘

總 分：一百分

※ 注意：不必抄題，作答時請將試題題號及答案依照順序寫在答卷上。 ( 於本試題紙上作答者，不予計分 )

１． Evaluate , where is the largest integer which is less than .

２． If a function is continuous on a bounded closed interval . Prove that takes on both a maximum value and a minimum value on .

３． If a function is differentiable on the open interval and continuous on the closed interval . Prove that there is at least one number in for which

.

４． Evaluate the indefinite integral

(1) (2)

５． Suppose that . Prove that .

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６． Find .

７． For each of the following statements, prove it if it is true, or give a counterexample if it is false.

(a) Let V and be vector spaces having same finite dimension, and let be a linear transformation. Then T is one-to-one if and only if

range( T ) = .

(b) Let A be a real matrix. Then A = 0 if and only if trace( where is the transpose of A .

８． Let .

(a) Find a matrix C such that is a diagonal matrix.

(b) Find .

９． Find the condition of k so that the linear system has exact one solution. In that case, find the solution.

１０． Let a, b and R , and let . What is the minimal polynomial for A ?

１１． Find the inverse of the matrix .

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