กระทู้นี้สำหรับคนชอบ Algebra แบบ advanced ครับ
ชุดนี้เอาโจทย์ linear algebra ที่เป็นข้อสอบ qualify มาลงครับ
ขอเป็นภาษาอังกฤษนะครับเพราะศัพท์เทคนิคเยอะเหลือเกิน
1. (UMCP August 2004) Let A be an invertible square matrix over C. Suppose A
n is diagonalizable for some n
ณ1. Show that A is diagonalizable.
2. (UMCP January 2004) Let A and B be nonzero nxn matrices over C. Suppose that AB=BA. Show that if the characteristic polynomial of A is separable(no multiple roots) then the minimal polynomial of B is separable.
3. (UMCP August 2003) Let V be a finite dimensional vector space over a field F. Let T:V-->V be a linear transformation.
(a) Suppose that every nonzero v
ฮV is an eigenvector of T. Show that T is a scalar multiple of the identity.
(b) A cyclic vector for T is a vector v
ฮV such that {v,Tv,T
2v,...} spans V. Suppose that every nonzero vector v
ฮV is a cyclic vector for T. Show that the characteristic polynomial of T must be irreducible over F.
(c) Suppose that V is 2-dimensional over F and that the characteristic polynomial of T is irreducible over F. Show that every nonzero v
ฮV is a cyclic vector for T.
4. (UMCP January 2003) Let M be an mxn matrix over a field F. Show that there exist an invertible mxm matrix A and invertible nxn matrix B such that AMB = (c
ij), with c
ii = 0 or 1 for all i and c
ij = 0 whenever i
นj.
5. (UMCP August 2002) Let A be an mxn matrix with real entries, where m
ณn. Assume A has rank n. Let x
ทy = x
Ty be the standard dot product on R
m.
(a) Show that A
TA is invertible. (Hint : Rewrite Ax
ทAx)
(b) Let P = A(A
TA)
-1A
T. Show that (x - Px)
ทPx = 0 for all x
ฮR
m