厦门大学高等代数考研试题集锦(2000-2013年).pdf

2000 ℄ 2 2 2 2 P [x]3 x + 1, x + x , x 1, x − x, x + x 1 + 2x + x2 (15%) A ∈ P m ×n , (A) = r  r B ∈ P m ×r r C ∈ P r ×n , Æ A = BC .(15%) x1 + x2 + · · · + xn = 0 M , x1 = x2 = · · · = xn N , Rn = M ⊕ N . (15%) A n V KerA = {α ∈ V | A (α) = 0} r r+1 , r , Æ KerA = KerA , s, KerA r = KerA r+s . (15%) A n A 2 = ε(). V Æ A Er 0 Er r 0 −En−r (15%) A n B , Æ 2 A = B .(15%) A, C n A C AX + XA + 2C = 0 X = B . B (10%) 1 2001 ℄ (15 ) XA = B  1 2 3  1 2 1   A = −1 3 1 , B = . 2 3 −2 0 6 5  1 1 1 1    1 1 −1 −1   (15 ) A = .   1 −1 1 −1 1 −1 −1 1 (1) A , (2) U , U AU (15 ) A, B, C ∈ F n ×n . (AB)+ (BC) ≤ (ABC )+ (B). (15 ) F [x]n F Æ n a , a , · · · , a f (x) = n (x − a ), i = 1, 2, · · · , n. 1 2 n i i=1,j= i j f (x), f (x), · · · , f (x) F [x] 1 2 n n (15 ) n