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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