課程名稱︰線性代數一
課程性質︰數學系必修
課程教師︰陳其誠
開課學院：理學院
開課系所︰數學系
考試日期︰2010年11月19日
考試時限：120分鐘
是否需發放獎勵金：是
(1) (25 points) Calculate A^(-1), where
┌ ┐
│2 0 3 -2 │
A = │1 1 1 0 │
│1 1 1 1 │
│1 0 2 0 │
└ ┘
(2) (25 points) Let β={v1,v2,v3,v4} be the ordered basis of R^4, with
v1=(2,1,1,1), v2=(0,1,1,0), v3=(3,1,1,2), v4=(-2,0,1,0).
Let T:R^4→R^4 be the linear transformation sending (x1,x2,x3,x4)∈R^4.
to (2x1+x3,x1+x2+x3+9x4,3x2+3x4,x1-x2+x3-x4)∈R^4. Calculate the matrix
[T]_β
(3) Either give a brief reason or give a counter example for each of the
following assertions (5 points each):
(a) If A is a 5*4 matrix and B is a 4*5 matrix, then the 5*5 matrix AB can
not be invertible.
(b) If A is a 5*4 matrix and B is a 4*5 matrix, then the 4*4 matrix BA can
not be invertible.
(c) If T,U:V→W are both linear and agree on a basis for V, then T=U.
(d) Every change of coordinate matrix is invertible.
(e) An n*n matrix having rank n is invertible.
(4) Give a rigorous proof for each of the following assertions.
(a) (15 points) Let V and W be finite-dimensional vector spaces over a
field F with ordered bases β and γ, respectively. For any linear
transformation T:V→W, the mapping T^t:W*→V* defined by T^t(g)=gT for
all g∈W* is a linear transformation with the property that
β* γ
[T^t] = ( [T] )^t
γ* β
(b) (10 points) Suppose A is a 5*5 matrix satisfying A^5=0 and A^4≠0.
(i) (5 points) Then A has rank 4 (Hint: There is some x∈F^5 such that
{x,Ax,(A^2)x,(A^3)x,(A^4)x}⊂F^5 is linearly independent. )
(ii) Then I-A is invertible (Hint: The kernal of L_(I-a) is trivial).
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