推 susanyen :€應該可以代替屬於 06/18 18:38
※ 編輯: princeeeeeee 來自: 140.112.249.177 (06/18 20:49)
→ princeeeeeee:已修改 06/18 20:49
課程名稱︰線性代數
課程性質︰必修
課程教師︰馮蟻剛(電機系四個老師考題都一樣)
開課學院：電資學院
開課系所︰電機系一年級
考試日期（年月日）︰2009 6/17
考試時限（分鐘）：100分鐘
是否需發放獎勵金：是
試題 :
Linear Algebra Final Examination
Dept. of Elec. Eng., National Taiwan University
June 17, 2009
USE OF ALL AUTOMATIC COMPUTING MACHINES IS PROHIBITED
1.Suppose the real nxn matrix A is invertible and diagonalizable, where n>1.
Briefly explain why the following matrices are all diagonalizable:
A^5+8A^2, A^(-1), A^T, (A^T)A,and A^~ obtained by 1) exchanging the
first and last rows of A and 2) exchanging the first and last columns of
the resultant matrix. (25%)
Four of these five matrices are always invertible and the other one may
become not invertible. Identify the one and explain why. (5%)
2.In the real space R^n, define the inner product<x,y>=x1y1+...+xnyx and the
norm ||x||=<x,x>^(1/2) for all vectors x=[x1 ... xn]^T and y=[y1 ... yn]^T.
Suppose S is a subspace of R^n. Let S^⊥ be the orthogonal complement of S
in R^n with respect to <‧,‧>. For any matrix A, let Null(A) denote its
null space. Consider any real 4x4 matrix A with Null(A) spanned by the set
{[2 0 2 -1]^T, [1 2 0 -1]^T,[3 -1 4 -1]^T}.
(a)Find an orthonormal basis for Null(A)^⊥.(10%)
(b)Find the vector z in Null(A) making ||z-[1 1 1 1 ]^T|| the smallest.(10%)
(Hint: It is easier to use the result of (a).)
(c)Find the set of all real 4x4 matrix A=A^T having the above Null(A).(10%)
(Hint: Recall the Spectral Decomposition Theorem.)
(註:下面以"€"代替"屬於"的那個數學符號因為我找不到orz)
(d)With the above Null(A), let any v€R^4 be uniquely decomposed as v=u+w,
where u€Null(A) and w€Null(A)^⊥, and define a linear operator T:R^4→R^4
as T(v)=u-w. Show that T is orthogonal. (10%) Also determine all
eigenvalues of T.(5%)
(Hint: Consider the proper equivalent condition for the orthogonality of T.)
3.Let V be the real vector spaces with vectors [p1(x) p2(x)]^T, where p1(x)
and p2(x) are polynomials in x with degrees no larger than 2. For vectors
[p1(x) p2(x)]^T and [q1(x) q2(x)]^T in V, [p1(x) p2(x)]^T+[q1(x) q2(x)]^T
=[p1(x)+q1(x) p2(x)+q2(x)]^T. For r€R,
r[p1(x) p2(x)]^T=[rp1(x) rp2(x)]^T.
(a)Find a basis B of V.(10%)
(b)Consider the linear operator D: V→V defined by D([p1(x) p2(x)]^T)
=[dp1(x)/dx dp2(x)/dx]^T. Find [D]B(註:B是下標) and decide if D is
one-to-one or onto.(15%)
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