課程名稱︰工程數學 - 線性代數
課程性質︰電機系必選
課程教師︰林茂昭
開課學院：電機資訊學院
開課系所︰電機工程學系
考試日期（年月日）︰2018/12/21
考試時限（分鐘）：1030-1210
Quiz 2 of Linear Algebra
1. (7%) Prove that if λ is an eigenvalue of an invertible matrix A,
then λ≠0 and 1/λ is an eigenvalue of A^-1.
2. Determine whether each of the following matrix is diagonalizable (using
real eigenvalues):
(a) (7%)
0 2 1
A = [ -2 0 -2]
0 0 -1
with characteristic polynomial -(t+1)(t^2+4).
(b) (7%)
-7 -3 -6
A = [ 0 -4 0]
3 3 2
with characteristic polynomial -(t+1)(t+4)^2/
3. Let A be an m× n matrix.
(a) (7%) Prove that A^{T}A and A have the same nill space.
(b) (7%) Prove that rank A^{T}A = rank A.
4. Let {w_1, w_2, ..., w_n} be an orthonormal basis of R^n.
Prove that for any vectors u and v in R^n,
(a) (8%) u‧v = (u‧w_1)(v‧w_1)+ ... +(u‧w_n)(u‧w_n).
(b) (7%) ∥u∥^2 = (u‧w_1)^2+ ... +(u‧w_n)^2
5. Let u = [-10 5]^T and W = span{[-3 4]^T}.
(a) (5%) Find the orthogonal projection matrix P_W.
(b) (5%) Obtain the unique vectors w in W and z in W^⊥ such that u = w + z.
(c) (5%) Find the distance from u to W.
6. (10%) Let
1 2 -1 -1
A = [-3 -5 2] and b = [ 0]. For the system of linear equation Ax=b,
2 3 -1 1
find the solution pf leat norm.
7. (10%) Let T ba a linear operator on R^n,
and suppose that {v_1, v_2, ..., v_n} is an orthonormal basis of R^n.
Prove that T is an orthogonal operator if and only if
{Tv_1, Tv_2, ..., Tv_n} is also an orthonormal basis for R^n.
8. (7%) Let A ba a symmetric n×n matrix with a spectral decompostion
A = μ_1P_1 + ... + μ_nP_n. Find a spectral decomposition of A^2.
9. (8%) Prove that for any matrix A,
the matrix A^{T}A is positive semidefinite.