中原大學九十學年度博士班入學招生考試

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科目：線性代數	（共1頁 第1頁）

1.	Matrix A's eigenvalues are -1,1 and 3,and the corresponding eigenvectors are (a) (10%)Find A. (b) (10%)Find the similarity transformation such that A is diagonalized . (c) (20%)Use(b)to evaluate (d) (10%)Which matrix function in(c) can be solved using the Cayley-Hamilton theorem? Show how.(You don't have to solve it again.) (e) (10%)Is A orthogonally diagonalizable? If yes,show how.
2.	(15%)Derive the least square solution to match the data points with the function
3.	(15%)Find a basis for the solution space of
4.	(10%)Let A, B, and Q be square matrices and B=QA. Show that the column space of A equals that of B if and only if Q is nonsingular.

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