推 trees880098 : 收電機系 05/27 00:42
課程名稱︰線性代數
課程性質︰系必修
課程教師︰蘇柏青
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰2015/5/21
考試時限（分鐘）：50 min
試題 :
Linear Algebra Quiz #2 Thursday May 21, 2015
1. Eigenvalues, Eigenvectors, and Diagonalization (30%)
_ _
| 3 -1 |
Let A = | |
| -1 3 |
- -
(a) (4%) Find the characteristic polynomial of A.
(b) (4%) Find eigenvalues of A. Indicate the multiplicity for each eigenvalue.
(c) (4%) For each of the eigenvalues of A, find a corresponding eigenvector.
(d) (4%) Find the eigenspace corresponding to each eigenvalue. Also indicate
the dimension of each of these eigenspaces.
(e) (6%) Is A diagonalizable? If so,find an invertible matrix P and a diagonal
matrix D that satisfy D = P^-1 A P.
(f) (4%) Is A orthoganally diagonalizable? That is, does there exist an
orthogonal matrix Q so that Q^T A Q is diagonal?
(g) (4%) Calculate A^6.
2. Gram - Schmidt Process (20%)
For each the following bases for R^3 , determine whether it is orthogonal,
orthonomal, or neither. If it is not orthonormal, apply the Gram-Schmidt
process and vector-length normalization to obtain an orthonormal basis.
(a) (5%) S1 = {[ 1 0 0 ]^T , [ 0 1 0 ]^T , [ 0 0 1 ]^T}.
(b) (5%) S2 = {[ 1 0 0 ]^T , [ 0 2 0 ]^T , [ 0 0 3 ]^T}.
(c) (5%) S3 = {[ (sqrt2)/2 (sqrt2)/2 0]^T,[(sqrt2)/2 -(sqrt2)/2 0^T],[0 0 3]^T}
(d) (5%) S4 = {[ 1 1 0 ]^T , [ 1 1 -1]^T , [ 1 1 -1]^T}.
3. Matrix representation of Linear Operators (50%)
Let T : R^3 -> R^3 be defined as
_ _ _ _
| x1 | | x1 + x3 |
T( | x2 | ) = | x2 - 2x3 |
| x3 | | 6x1 - x2 + 3x3 |
- - - -
_ _ _ _ _ _
| 0 | | 1 | | 1 |
and let B = {| -1 | , | 0 | , | -1 | } be a basis for R^3.
| 1 | | -1 | | 1 |
- - - - - -
(a) (2%) Find A , the standard matrix of T.
_ _
| 1 |
(b) (9%) Let v = | 0 | . Calculate [v] , T(v) and [T(v)] .
| -1 | B B
- -
(c) (9%) Find [T] , the matrix representation of T with respect to B.
B
(d) (5%) Determine whether 5 is an eigenvalue of A by claculating det( A - 5I).
If so , find the eigenspace of A corresponding to the eigenvalue 5.
(e) (5%) Find the characteristic polynomial for A.
(f) (5%) Is A diagonalizable? If so, find an invertible matrix P and a diagonal
matrix D1 such that A = P D1 P^-1.
(g) (5%) Determine whether 1 is an eigenvalue of [T] by calculating det([T]-5I)
B B
If so, find the eigenspace of [T] corresponding to the eigenvalue 5.
B
(h) (5%) Find the charateristic polynomial for [T] .
B
(i) (5%) Is [T] diagonalizable?If so,find an invertible matrix Q and a diagonal
B
matrix D2 such that [T] = Q D2 Q^-1.
B
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