國立彰化師範大學 95 學年度碩士班招生考試試題
系所： 數學系
科目： 線性代數
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1. Let T : R 3  R 3 be a linear transformation defined by
T ( x, y, z )  ( x  y, y  z, z  x) for all x, y, z  R ,
where R is the set of real numbers.
(1) Find the matrix representation [T ] of T with respect to the standard basis
  {(1,0,0), (0,1,0), (0,0,1)} . (3％)
(2) Find the matrix representation [T ] of T with respect to the basis
  {(1,1,0), (1,0,1), (0,1,2)} . (7％)
(3) Find an invertible matrix P such that [T ]  P[T ] P1 . (5％)
0  2 
2. Let A  
.
1 3 
(1) Compute An in terms of n . (12％)
(2) Can we find an orthogonal matrix P over real numbers and a diagonal matrix D such
that PAP1  D ? Please explain the reason. (5％)
3. Let W  {( x, y, z ) : x  2 y  z  0} be a subspace of R 3 .
(1) Find an orthonormal basis of W . (5％)
(2) Find the orthogonal projection of the vector (1,1,1) on W . (5％)
4. Let A be a m n matrix and let At denote the transpose of A .
(1) Suppose that A is a matrix over real numbers. Prove that rank( At A ) = rank( A ).(8
％)
(2) Suppose that A is a matrix over complex numbers. Prove or disprove that rank( At A )
= rank( A ).(7％)
5. Let W1 and W2 be two subspaces of a finite dimensional vector space V . Let W1  W2
be the set {w1  w2 : for all w1  W1 and w2 W2} . Prove that
(1) W1 W2 and W1  W2 are subspaces of V ; (6％)
(2) dim( W1  W2 ) = dim W1 +dim W2 -dim( W1 W2 ).(12％)
6. Let A be an n n matrix over real numbers and let At denote the transpose of A .
Suppose that A is a skew symmetric matrix, that is At   A . Show that
(1) every eigenvalue of A is zero or purely imaginary; (5％)
(2) if n is an odd number, then the determinant of A is zero. (5％)
7. Find all possible Jordan forms for a 6 6 matrix over complex numbers with
characteristic polynomial ( x  1)( x  2) 2 ( x  3)3 . (15％)
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