課程名稱︰線性代數
課程性質︰系必修
課程教師︰馮世邁
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰99年6月3日
考試時限（分鐘）：50 min
是否需發放獎勵金：是!感謝!
試題 :
1.Let A and u be defined below. Let W=Col A.
┌ ┐ ┌ ┐
│ 1 1 1 1 │ │ 1 │
A = │ 0 1 -1 -1 │ , u= │ 0 │
│ 1 1 1 1 │ │-1 │
│ 0 1 -1 2 │ │ 2 │
└ ┘ └ ┘
(a) (20%) Find an orthonormal basis Β fow W.
(b) (10%) Extend Β to form an orthonormal basis for R^4.
(c) (5%) Find P_W.
(d) (5%) Find the orthogonal projection of u onto W.
┴
(e) (5%) Find the orthogonal projection of u onto W .
2.
┌ ┐
│ 4 2 2 │
A = │ 2 4 2 │
│ 2 2 4 │
└ ┘
(a) (25%) Find orthogonal P and diagonal D such that A=PDP^T
(b) (5%) Find the spectral decomposition of A.(You may express your answer in
the form of u_i u_i^T.)
(c) (5%) Is A invertible? If it is, find the spectral decomposition of A^-1.
3. (10%) Show that 2 eigenvectors corresponding to distinct eigenvalues of a
symmetric matrix are orthogonal.
4.(10%) Let S_1 ={u_1, u_2, ......,u_k } be a linearly independent subset of
R^n
and let u be a linear combination of vectors in S_1.
Let S_2 ={u_1, u_2, ......,u_k, u}. Suppose we apply the Gram-Schimdt
process to S_1 and S_2 to obtain the following orthogonal sets:
S'_1 ={v_1, v_2, ......, v_k},
S'_2 ={v_1, v_2, ......, v_k, v}.
What is v? (Explain your answer)
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.4.196