Solutions for Math 225 Assignment #4 () Let B {(3, 4), (4, 5)} and C {(, ), (0, )} be two ordered bases of R 2 (a) Find the change-of-basis matrices P C B and P B C (b) Find v] B if v] C ] (c) Find v] C if v] B ] Solution Let A {e, e 2 } be the standard basis Then ] ] 3 4 0 P A B and P 4 5 A C Then P C B P C A P A B P ] ] 0 3 4 4 5 A C P A B 0 and ] 3 4 P B C P C B For v satisfying v] C, ] ] ] 3 4 4 5 ] 4 3 ] 4 v] B P B C v] C 3] For v satisfying v] B, ] v] C P C B v] B ] 3 4 ] ] 5 4 ] 0 ] 3 4 http://wwwmathualbertaca/ xichen/math2254f/hw4solpdf
2 (2) Let V be the subspace of Rx] consisting of polynomials of degree 3 Let B {, x, (x ) 2, (x ) 3 } and C {, x +, (x + ) 2, (x + ) 3 } be two ordered bases of V (a) Find the change-of-basis matrices P C B and P B C (b) Find v] B if v] C 0 0 (c) Find v] C if v] B 0 0 Solution Since x 2 + (x + ) (x ) 2 ( 2 + (x + )) 2 4 4(x + ) + (x + ) 2 (x ) 3 ( 2 + (x + )) 3 8 + 2(x + ) 6(x + ) 2 + (x + ) 3 P C B 2 4 8 ] ] C x ] C (x ) 2 ] C (x ) 3 ] C 0 4 2 0 0 6 0 0 0 Since x + 2 + (x ) (x + ) 2 (2 + (x )) 2 4 + 4(x ) + (x ) 2 (x + ) 3 (2 + (x )) 3 8 + 2(x ) + 6(x ) 2 + (x ) 3 P B C 2 4 8 ] ] B x + ] B (x + ) 2 ] B (x + ) 3 ] B 0 4 2 0 0 6 0 0 0
For v satisfying v] C 0 0, 2 4 8 9 v] B P B C v] C 0 4 2 0 0 0 6 0 2 6 0 0 0 For v satisfying v] B 0 0, 2 4 8 7 v] C P C B v] B 0 4 2 0 0 0 6 0 2 6 0 0 0 3 (3) Which of the following statements are true and which are false? Justify your answer (a) For every 2 2 matrix A, {I, A, A 2 } is linearly dependent Proof True Let f(x) det(xi A) x 2 + r x + r 2 be the characteristic polynomial of A Then f(a) A 2 + r A + r 2 I 0 by Cayley-Hamilton Theorem So I, A, A 2 are linearly dependent (b) Let B {v, v 2,, v n } and C {w, w 2,, w n } be two ordered bases of a vector space V If v] B v] C for all v V, then B C Proof True For v v k ( k n), v k ] B e k Since v k ] B v k ] C, v k ] C e k Therefore, v k 0 w + + 0 w k + w k + 0 w k+ + + 0 w n w k for k, 2,, n So B C
4 (c) Let v, v 2,, v n be n vectors in a vector space V If V Span{v, v 2,, v n }, then dim V n Proof False For example, let V R, v 0 and v 2 Then V Span{v, v 2 } but dim V (d) Let V and V 2 be two subspaces of a vector space V If B is a basis of V and B 2 is a basis of V 2, B B 2 is a basis of V V 2 Proof False For example, let V V 2 V R, B {} and B 2 {2} Then B is a basis of V and B 2 is a basis of V 2 but B B 2 is not a basis of V V 2 R (4) Let A be a square matrix satisfying A n 0 for some some positive integer n Show that I, A, A 2,, A n are linearly independent if and only if A n 0 Proof Suppose that I, A, A 2,, A n are linearly independent If A n 0, then 0 I + 0 A + + 0 A n 2 + A n 0 and hence I, A, A 2,, A n are linearly dependent Contradiction So A n 0 if I, A, A 2,, A n are linearly independent Suppose that A n 0 Let c 0, c,, c n be constants such that c 0 I + c A + + c n A n 0 It suffices to show that c 0 c c n 0 We prove c k 0 by induction on k First, multiplying the above identity by A n on both sides, we obtain A n (c 0 I + c A + + c n A n ) 0 c 0 A n + c A n + + c n A 2n 2 0 Since A n 0, A m 0 for all m n So c 0 A n 0 And since A n 0, c 0 0 Second, suppose that c 0 c c k 0 Then c k+ A k+ + c k+2 A k+2 + + c n A n 0
Multiplying the above identity by A n k 2 on both sides, we obtain A n k 2 (c k+ A k+ + c k+2 A k+2 + + c n A n ) 0 c k+ A n + c k+2 A n + + c n A 2n k 2 0 It follows that c k+ A n 0 and hence c k+ 0 This shows that c 0 c c n 0 and hence I, A, A 2,, A n are linearly independent if A n 0 (5) Which of the following maps are linear transformations and which are not? Justify your answer (a) T : R 2 R 3 given by T (x, y) (x y, y, 2x + y); 5 Proof It is a linear transformation because T (x + cx 2, y + cy 2 ) ((x + cx 2 ) (y + cy 2 ), y + cy 2, 2(x + cx 2 ) + (y + cy 2 )) (x y, y, 2x + y ) + c(x 2, y 2, 2x 2 + y 2 ) T (x, y ) + ct (x 2, y 2 ) (b) T : M n n (R) M n n (R) given by T (A) 2A + A T ; Proof It is a linear transformation since T (A + cb) 2(A + cb) + (A + cb) T 2A + 2cB + A T + cb T (2A + A T ) + c(2b + B T ) T (A) + ct (B) (c) T : Rx] R given by T (f(x)) 3 f()f(2)f(3); Proof It is not a linear transformation since T (), T (x) 3 6 and T ( + x) 3 24 + 3 6 T () + T (x) (d) T : Rx] Rx] given by T (f(x)) x 2 f (x) + f(x 3 )
6 Proof It is a linear transformation since T (f(x) + cg(x)) x 2 (f(x) + cg(x)) + (f(x 3 ) + cg(x 3 )) x 2 (f (x) + cg (x)) + (f(x 3 ) + cg(x 3 )) (x 2 f (x) + f(x 3 )) + c(x 2 g (x) + g(x 3 )) T (f(x)) + ct (g(x)) (6) Let P 3 be the subspace of Rx] consisting of all polynomials of degree 3 Let B {(x )(x 2)(x 3), x(x 2)(x 3), x(x )(x 3), x(x )(x 2)} (a) Show that B is a basis of P 3 (b) Find + x + x 2 ] B (c) Find f(x) in P 3 satisfying that f(0), f(), f(2) 0 and f(3) 2 Proof (a) Let f 0 (x) (x )(x 2)(x 3) f (x) x(x 2)(x 3) f 2 (x) x(x )(x 3) f 3 (x) x(x )(x 2) Note that f 0 (0) f (0) f 2 (0) f 3 (0) 6 A f 0 () f () f 2 () f 3 () f 0 (2) f (2) f 2 (2) f 3 (2) f 0 (3) f (3) f 2 (3) f 3 (3) 2 2 6 If c 0 f 0 (x) + c f (x) + c 2 f 2 (x) + c 3 f 3 (x) 0 for some constants c 0, c, c 2, c 3, then c 0 f 0 (0) + c f (0) + c 2 f 2 (0) + c 3 f 3 (0) 0 c 0 0 c 0 f 0 () + c f () + c 2 f 2 () + c 3 f 3 () 0 A c c 0 f 0 (2) + c f (2) + c 2 f 2 (2) + c 3 f 3 (2) 0 c 2 0 0 c 0 f 0 (3) + c f (3) + c 2 f 2 (3) + c 3 f 3 (3) 0 c 3 0 And since A is nonsingular, c 0 c c 2 c 3 0 So B {f 0 (x), f (x), f 2 (x), f 3 (x)} is linearly independent And since dim P 3 4, B is a basis
(b) Let g(x) x 2 + x + and Then g(x)] B x 2 + x + ] B a 0 a a 2 a 3 a 0 f 0 (x) + a f (x) + a 2 f 2 (x) + a 3 f 3 (x) g(x) and a 0 f 0 (0) + a f (0) + a 2 f 2 (0) + a 3 f 3 (0) g(0) a 0 f 0 () + a f () + a 2 f 2 () + a 3 f 3 () g() A a 0 f 0 (2) + a f (2) + a 2 f 2 (2) + a 3 f 3 (2) g(2) a 0 f 0 (3) + a f (3) + a 2 f 2 (3) + a 3 f 3 (3) g(3) a 0 /6 x 2 + x + ] B a a 2 3/2 7/2 3/6 a 3 (c) Suppose that a 0 a a 2 a 3 f(x) b 0 f 0 (x) + b f (x) + b 2 f 2 (x) + b 3 f 3 (x) g(0) g() g(2) g(3) for some constants b 0, b, b 2, b 3 By the same argument as above, we obtain b 0 f(0) b 0 /6 A b b 2 f() f(2) 0 b b 2 /2 0 f(3) 2 /3 So b 3 f(x) 6 (x )(x 2)(x 3) 2 x(x 2)(x 3) + x(x )(x 2) 3 (7) Let B be an ordered basis of a vector space V over R of dimension n Do the following: (a) Show that b 3 c v + c 2 v 2 ] B c v ] B + c 2 v 2 ] B for all v, v 2 V and c, c 2 R (b) Let T : V R n be the map given by T (v) v] B Show that T is a linear transformation 7
8 Proof Let B {w, w 2,, w n } Suppose that Then and hence a b a v ] B 2 and v b 2] B 2 a n b n v a w + a 2 w 2 + + a n w n v 2 b w + b 2 w 2 + + b n w n c v + c 2 v 2 (c a + c 2 b )w + (c a 2 + c 2 b 2 )w 2 + + (c a n + c 2 b n )w n Therefore, c a + c 2 b a b c c v +c 2 v 2 ] B a 2 + c 2 b 2 c a 2 +c b 2 2 c v ] B +c 2 v 2 ] B c a n + c 2 b n a n b n Let c and c 2 c Then T (v + cv 2 ) v + cv 2 ] B v ] B + cv 2 ] B T (v ) + ct (v 2 ) for all v, v 2 V and c R So T is a linear transformation (8) Let B {v, v 2,, v n } be an ordered basis of R n, A be an n n invertible matrix and C {Av, Av 2,, Av n } Find P C B and P B C Express your answer in terms of v, v 2,, v n and A and justify it Solution Let D {e, e 2,, e n } be the standard basis of R n Then P D B v ] B v 2 ] B v n ] B ] v v 2 v n ] P P D C Av ] B Av 2 ] B Av n ] B ] Av Av 2 Av n ] AP Therefore, P C B P C D P D B P D C P D B (AP ) P P AP P B C P C B (P AP ) P A P