課程名稱︰線性代數 課程性質︰必修 課程教師︰鄭振牟 開課學院：管理學院 開課系所︰工商管理學系科技管理組 考試日期（年月日）︰2009.12.04 考試時限（分鐘）：100 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : (以下以€代替屬於) 1. (10%) Suppose P€L(V) and P^2＝P. Prove that V＝nullP⊕rangeP. 2. For each of the following, give an example operator as described. (a) (5%) Give an example of an operator whose matrix with respect to some basis conains only nonzero numbers on the diagonal, but the operator is not invertible. (b) (5%) Give an example of an operator whose matrix with respect to some basis contains only 0's on the diagonal, but the operator is invertible. 3. (10%) Suppose T€L(V) and dim rangeT＝k. Prove that T has at most k+1 distinct eigenvalues. 4. (10%) Define T€L(F^3) by T(z ,z ,z )＝(2z ,0,5z ). 1 2 3 1 2 Find all eigenvalues and eigenvectors of T. 5. (10%) Prove that if U is a subspace of V such that U≠{0} and U≠V, then there exists T€L(V) such that U is not invariant under T. 6. (10%) Suppose T€L(F^n,F^m) amd that ┌ a ... a ┐ │ 1,1 1,n │ M(T)＝│ : ... : │ , │ a a │ └ m,1 ... m,n ┘ where we rae using the standard bases, Prove that M(Tv)＝M(T)M(v) by showing that T(x ,...,x )＝(a x +...+a x ,...,a x +...+a x ) 1 n 1,1 1 1,n n m,1 1 m,n n for every (x ,...,x )€F^n. 1 n 7. (10%) Suppose that U and V are finite-dimensional vector spaces and that S€L(V,W), T€(U,V). Prove that dim nullST≦dim nullS + dim nullT. 8. (10%) Suppose that V and W are finite dimensional and that U is a subspace of V. Prove that there exists T€L(V,W) such that nullT＝U if amd only if dim U≧dim V-dim W. 9. (10%) Suppose that V is finite dimensional and that T€L(V,W). Prove that there exists a subspace U of V such that U∩nullT＝{0} and rangeT＝ {Tu:u€U}. 10. Let U and U be two subspaces of R^2. 1 2 (a) (5%) Give an example where U ∪U is not a subspaceof R^2. 1 2 (b) (5%) Suppose U ⊕U ＝R^2. Give an example where U '≠U but U '⊕U ＝ 1 2 1 1 1 2 R^2. --

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