課程名稱︰工程數學-線性代數
課程性質︰電機系必修
課程教師︰劉志文
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰109.01.10
考試時限（分鐘）：100
試題 :
1.(10%) Probe that any orthogonal subset of R^n consisting of nonzero vector
is linearly independent.
2.(25%) The "Orthogonal Decomposition Theorem" is stated as follows.
Let W be a subspace of R^n. Then, for any vector n in R^n, there exists unique
vector w in W and z in W perp such that u = w + z. In addition, if {v1, v2, ..
., vk} is an orthonormal basis for W, then w = (u‧v1)v1 + (u‧v2)v2 + ... +
(u‧vk)vk.
(A)(5%) Show that w=Pw u, where Pw is the orthogonal projection matrix on W
and Pw = v1v1^perp + v2v2^perp + ... + vkvk^perp.
(B)(5%) Define vivi^perp = Pi, prove that rank(Pi) = 1, PiPi = Pi, for i = 1
..., k, and PiPj = 0 (zero matrix) for i is not equal to j.
(C)(5%) Prove that Pw = P1 + P2 + ... + Pk is a symmetric matrix and Pw^2 = Pw
(D)(5%) Find the eigenvalues of Pw and their corresponding eigenspaces.
(E)(5%) Fint the orthogonal projection matrix on W perp, P_{w perp}, in temrs
of Pw and In which is nxn identity matrix.
3.(15%) Let T be the linear operator on R^n defined by
┌x1┐ ┌-x1 ┐
│x2│ │ x2 │
T(│x3│) = │ x3 │
│. │ │ . │
│. │ │ . │
│. │ │ . │
└xn┘ └ xn ┘
(A)(5%) Show that T is an orthogonal operator.
(B)(5%) Show that the only eigenvalues of T are 1 and -1, respectively.
(C)(5%) Let W be the eigenspaces of T corresponding to eigenvalue 1. Let Q be
the standard matrix of T, and Pw be the orthogonal projection matrix on W.
Show that Q = 2Pw - In, where In is nxn identity matrix.
4.(12%) Let
┌-5 6 1┐
A = │-1 2 1│
└-8 6 4┘
where eigenvalues are 2, 3, and -4.
(A)(6%) Please find the eigenvalues of A^-1.
(B)(6%) Please find the eigenvalues of A^2 + 2A.
5.(8%) A square matrix A is a nilpotent matrix if there exists a positive int-
eger p such that A^p = O. Show that the only eigenvalue of A is 0.
6.(10%) Determine all values of a such that the following matrix is NOT diago-
nalizable.
┌-1 0 0┐
│ 3 -2 1│
└ 0 0 a┘
7.(5%) In the vector space of real-valued function F = {f│f:R→R}, determine
if the following set S is linearly independent.
S = {sin^2x, cos^2x, 2}
8.(10%) Let M_2x2 be the set of all 2x2 matrices. Let T be the function on M
defined by T(A) = (traceA)┌1 2┐. If A = ┌a11 a12┐, trace A = a11 + a12.
└3 4┘ └a21 a22┘
(A)(5%) Prove that T is linear.
(B)(5%) Supposed that B is the basis for M,
B = {┌1 0┐,┌0 1┐,┌0 0┐,┌0 0┐}.
└0 0┘ └0 0┘ └1 0┘ └0 1┘
Determine [T]_B.
9.(5%) Let L(V,W) be the set of all linear transformations from vector space V
to vector space W. Suppose V is the set of all mxn matrices, V = M_mxn and W
is the set of all linear transformations from R^n to R^m, W = L(R^n,R^m).
Please find the dimension of L(V,W).