課程名稱︰線性代數一
課程性質︰數學系大一必修
課程教師︰莊武諺
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2018.11.7
考試時限（分鐘）：150
試題 :
以下的ε均代表屬於
滿分115分
(1)(15 points) Let T: R^2→R^3 be the linear transformation defined by
T((1,3))=(2,2,2) and T((3,1))=(1,-3,0).
Letα={(-2,2),(2,2)}andβ={(2,3,1),(1,2,1),(1,1,1)}
be ordered bases of R^2 and R^3 respectively.
Find the matrix representation β
[T]
α
(2)(15 points) Let T: R^5→R^3 be the linear transformation given by T(x)=Ax,
1 -2 3 2 1
where A=(4 2 3 -3 1)
2 0 0 2 1
Please find bases of the null sapce N(T) and the range R(T).
(3)(15 points) Let V be a finite-dimensional vector space and T:V→V be a
linear transformation. Suppose that rank(T)=rank(T^2).
Prove that V=R(T)⊕N(T).
(4)(15 points)
(a) Let A εMnxn(F) satisfying A^7=0. Show that In-A is invertible
and compute (In-A)^-1.
(b) Let A,B εMnxn(F). Suppose that A+B is invertible. Show that
A(A+B)^-1B=B(A+B)^-1A.
(5)(15 points) Let A be an invertible upper-triangular nxn matrix.
Show that A^-1 is also upper-triangular,
(6)(20 points) Let A,B,C,D εMnxn(F) and OεMnxn(F) be the zero matrix.
A O
(a) Show that det(C D)=det(A)det(D).
(b)Further assume that D is invertible and CD=DC.
A B
Show that det(C D)=det(AD-BC).
(7)(20 points) Let V be an m-dimensional vector space over an infinite field F
and{u1,...,un}be a linearly independent subset of V. Prove that
for any v1,...,vnεV,{u1+αv1,...,un+αvn}is linearly
independent over F for all but finitelymany values αεF.
(Hint: You can only prove the statement for the m=n case to get
partial points. Feel free to apply the fact that a degree d
polynomial with coefficients in F has at most d zeros in F.)