國立新竹教育大學九十八學年度研究所碩士班招生考試試題
所別：應用數學系碩士班
科目：線性代數（本科總分：150 分）
※ 請橫書作答
1 1 
 1 1
1 0 
 0 1
1. Let A1  
 , A2  
 , A3  
 and A4  
.
1 0 
 0 1
1 1 
 1 1
(a) Prove that A1 , A2 , A3 and A4 form a basis for M 22 () . (10 分)
a
a 
(b) Find the unique representation of arbitrary vector  1 2  in
 a3 a4 
M 22 () as a linear combination of A1 , A2 , A3 and A4 . (10 分)
(c) What is the dimension of M 22 () ? (5 分)
2. Let c0 , c1 and c2 be three distinct scalars in an infinite field F .
(a) Describe the Lagrange interpolation polynomial f associated with
c0 , c1 and c2 . (10分)
(b) Use the Lagrange interpolation formula to construct the
polynomial of smallest degree whose graph contains the points
(-4,24), (1,9) and (3,3). (10分)
3
3. Determine whether the following sets are subspaces of  under the
operations of addition and scalar multiplication defined on 3 .
Justify your answers.
(a) W1  (a1 , a2 , a3 ) 3 : a1  4a2  a3  0 . (10 分)
(b) W2  (a1 , a2 , a3 ) 3 : 5a12  3a2 2  6a32  0 . (10 分)
 5  1 0
4. Let A   1 5 0 .
 0 0 4
(a) Find the eigenvalues of A . (5 分)
(b) Find the algebraic and geometric multiplicity with respect to the
eigenvalues for A . (5 分)
(c) Find an orthogonal matrix P such that P 1 AP is diagonal.
(10 分)
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1  3
3

5. Let A   7  2 9  .
 2  1 4 
(a) Find the eigenvalues of A . (5 分)
(b) Find the algebraic and geometric multiplicity with respect to the
eigenvalues for A . (5 分)
(c) Find the Jordan canonical form of A . (10 分)
6. Let T :  2   2 be the linear transformation defined by
T ( x, y )  ( x  2 y , 2 x  y )
Find an orthonormal basis of eigenvectors of T for  2 . (10 分)
7. Prove or disprove:
If a matrix A is positive definite, then the eigenvalues of A is
positive. (10 分)
8. Find a normalized QR  decomposition for A and use it to solve the
least-squares problem Ax  y , where
1 2 
 2


A  2 1  and y  3
1  1
0
(15 分)
9. Let V be an inner product space and u, v  V . Prove that
 u, v   u v (the Cauchy-Schwarz inequality). (10 分)
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