課程名稱︰應用代數
課程性質︰選修
課程教師︰呂學一
開課學院：電機資訊
開課系所︰資訊工程
考試日期（年月日）︰2015/06/26
考試時限（分鐘）：180
試題 :
總共十一題，每題十分，可按任何順序答題。只能參考個人事先準備的A4單頁大抄。前五
題都是一個可能對也可能不對的敘述。如果你覺得對，請證明它是對的，如果你覺得不對
，請證明它是錯的。課堂上證過的定理，或是提過的練習題，都可以直接引用。
第一題
Let S,S',S" be coordinate systems with coincided x-axes, parallel y-axes, and
parallel z-axes. At time 0 with respect to all three clocks, S,S',S" coincide.
Suppose that system S"(respectively, S" and S") moves at a postive velocity v_1
(respectively, v_2 and v_3) measured by S(respectively, S' and S). Thus, the
correspondence of coordinates from S(respectively,S' and S) to S'(respectively,
S" and S") is described by T_(v_1) (respectively,T_(v_2) and T_(v_3). If
T_(v_3) = T_(v_2)T_(v1), then
v_1 + v_2
v_3 = ------------
1 + v_1v_2
第二題
Find a generalized eigenbasis of M_(2x2)(C) for T∈L( M_(2x2)(C),M_(2x2)(C))
with T(A) = 2A+A^t for each A∈M_(2x2)(C)
第三題
Let T∈L(V,V) for vector space V with dim(V) < ∞. If βis a cycle of
generalized eigenvectors of T corresponding to eigenvalue λof T, then span(β)
is a T-invariant subspace of V.
第四題
Let T∈L(V,V) for vector space V with dim(V) < ∞ over C. Let λ_1,...,λ_k be
the distinct eigenvalues of T. If T is diagonalizable, then
rank( T-(λ_i)(I_v) ) = rank( (T-(λ_i)(I_v))^2 )
holds for each i = i,...,k
第五題
If J is an n×n real matrix in Jordan canonical form,then J and J^t are similar.
第六題
Find a Jordan canonical form of
┌ ┐
│ 3 -1 -1 │
│ 1 0 -1 │
│-2 5 4 │
└ ┘
第七題
Determine which of the following four matrices A,B,C,D are similar:
┌ ┐ ┌ ┐
│ 3 -1 -1 │ │ 0 1 2 │
A= │ 1 0 -1 │ , B= │ 0 1 1 │ ,
│-2 5 4 │ │ 0 0 2 │
└ ┘ └ ┘
┌ ┐ ┌ ┐
│ 0 -3 7 │ │ 0 1 -1 │
C= │-1 -1 5 │ , D= │-4 4 -2 │
│-1 -2 6 │ │-2 1 1 │
└ ┘ └ ┘
第八題
Find the minimal polynomial of
┌ ┐
│ 3 0 1 │
│ 1 0 -1 │
│ 0 0 2 │
└ ┘
第九題
Let T∈L(V,V) for vector space V with dim(V) < ∞. Prove that if U is a
T-invariant subspace of V, then f_(T_U) | f_T(t).
第十題
Let T∈L(V,V) for vector space V with dim(V) < ∞. Let U be the T-cyclic
subspace of V generated by a nonzero vector x∈V with dim(U) = k. Prove both of
the following statements:
k-1
1. {v,T(v),...,T (v)} is a basis of U
k k-1
2. -T (v) = a_0 v + a_1 T(v)+...+a_(k-1) T (v) implies
k k-1 k
(-1) f_(T_U) (t) = a_0 + a_1t +...+a_(k-1) t +t .
第十一題
Prove Cayley-Hamilton Theorem:
f_T(T) = T_0. (Your proof may directly use the statements of Problems 9 and 10.)
--
※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 140.112.251.92
※ 文章網址: https://www.ptt.cc/bbs/NTU-Exam/M.1435337668.A.788.html
推
06/27 01:08, , 1F
06/27 01:08, 1F