课程名称︰线性代数一
课程性质︰数学系大一必修
课程教师︰朱桦
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰103.11.19
考试时限（分钟）：140min
P.S. A*代表共轭 γ=gamma
试题 :
(1) (30%) In the following "vector space" V over F with the usual operation,
which of the them are isomorphic?
(a) V1=C ; F=R
(b) V2 is a vector space with 25 elements ;F=F_5
(c) V3=C^3/U,where U={(az,a,aw):a∈C},z,w are not all zero; F=C
(d) V4=P(C)/{f∈P(C):f=0 or deg(f)>=2}, P(C) is the space of all polynomal; F=C
(e) V5={f∈P(R): f''(x)=0} ;F=R
(f) V6={A∈M_2x2(C):A*=-A} ,Where [A11 A12]*=[A11* A12*]; F=R
[A12 A22] [A21* A22*]
(g) V7={A∈M_2X2(C):A11-A12+A21-A22=0}; F=R
(h) V8=N(T),where the linear transformation T:F^3->F^3 is defined by
T(x1,x2,x3)=(x1+2x2+3x3,3x1+3x2+x3,2x1+x2+3x3);F=F_5
(i) V9=R(T),where T is defined as in (h); F=F_5
(J) V10=W1+W2,where
W1=span({(1,2,3,4+i,0),(0,i,2,1,4),(3-i,2,0,0,0),(4,2i,1,0,0)})
W2=span({(2,3i,0,0,0),(0,i,0,4,0),(3,0,1,0,0),(0,0,1,0,-i)});F=C
(2) (10%) Find a basis for the solution space of the system of linear equations
2x1+ x2-4x3 +8x5=0
3x1 -6x3+2x4+ x5=0
2x2 + x4 =0
4x2 +3x4-4x5=0
-4x1+3x2+8x3-2x4+2x5=0
(3) (10%) Construct a polynomial of smallest degree whose graph contains the
points (-2,0),(-1,6),(0,4),(1,6),(2,24)
(4) (10%) Let S={ [1 0],[0 -1],[-1 2],[2 1] }
{ [-2 1] [1 1] [ 1 0] [2 -2] }
Determine whether the set S is linearly independent in the space
(a)M_2X2(R) (b)M_2x2(F_3)
(5) (10%) Let αbe the standard basis for P_2(R),β={x^2,2x+2x^2,2+x},and
γ={-1,3+x,2-x+x^2}
Let T be a linear operator on P_2(R) defined by :
x
T(f(x))=(x+1)f'(x)-∫f(t)dt/x
0
(a)Find the matrix that change α-coordinates into γ-coordinates.
γ
(b)Find[T]
β
(6) (10%) Let T be a linear operator on a vector space V over R such that T^2=Iv
Let U={v∈V:T(v)=v},and W={v∈V:T(v)=-v}.Show that V＝U⊕W
(7) (10%) Let V and W be vector spaces,and let T and U be nonzero linear
transformation from V into W.
If R(T)∩R(U)={0},Show that T and U are linearly independent.
(8) (10%) Let V be an n-dimensional vector space, and let T be a linear
operator on V such that T^2=T.
Show that there is a basis β for V such that [T] has the form:
β
[I_k O]
[O O] for some k<=n
(9) (10%) Let V and W be vector spaces such that dim(V)=dim(W),and let T:V->W be linear.
Show that there exist bases βand γfor V and W,respectively,such that
γ
[T] has the form [I_k O]
β [O O] for some k<=n
(10) (10%) Let W be the subspace of M_nxn(F) spanned by the matrices C of the
form:C=AB-BA.
Show that W is the subspace of matrices which have trace zero