Mathematics 206 Solutions for HWK 22b Section 8.4 p399

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1 Mathematics Solutions f HWK b Section 8. p99 Problem, 8. p99. Let T : P P be the linear transfmation defined by T(p(x)) = xp(x). (a) Find the matrix f T with respect to the standard bases B = {u, u, u }, B = {v, v, v, v } taken in the der u (x) =, u (x) = x, u (x) = x v (x) =, v (x) = x, v (x) = x, v (x) = x (b) Verify that this matrix functions as it should. In other wds, verify that multiplying the B-codinate matrix f p on the left by this matrix always produces the B -codinate matrix f T(p). Solution. (a) Since T() = x, T(x) = x, T(x ) = x, we have T(u ) = v, T(u ) = v, T(u ) = v. Therefe the matrix f T relative to B and B is A = T(u ) B T(u ) B T(u ) B = (b). With p given by p(x) = c + c x + c x, we should have T(p) B = Ap B Since T(p) is the function c x + c x + c x, we should have A c c c = c c c c c c c = c c Page of 9 A. Sontag April 9,

2 Math HWK b Solns contd 8. p99 which is exactly right. Problem 5, 8. p99. Let T: R R be the linear transfmation defined by T( x x ) = x + x x (a) Find the matrix f T relative to the basis B = {u, u } f R and B = {v, v, v } f R, given that u =, u =, v =, v =, v = (b) Verify that this matrix functions as it should. In other wds, verify that multiplying the B-codinate matrix f v on the left by this matrix always produces the B -codinate matrix f T(v). Solution. (a). T(u ) = T( ) = 7 = + 8 = v v + 8 v so so T(u ) = T( T(u ) B = ) = 8 = v + v + v T(u ) B = Page of 9 A. Sontag April 9,

3 Math HWK b Solns contd 8. p99 Therefe the matrix f T relative to B and B is T B,B = = 8 (b) We want to verify that f every u R we have With u = au + bu we would have T(u) B = T(au + bu ) B = T(a 8 T(u) B = u B 8 + b a b ) B = T( ) = a + b B 7a + b a + b B and so it should be that which does hold. u B = 8 8 a b = a + b a + 8b 7a + b (v + ( a + b)v + (a + 8b)v ) = a + b 7a + b ( a + b) + (a + 8b) = a + b a + b a + b + a + b a + b + 8a + b 8a + b = 7a + b = a + b 7a + b a + b Page of 9 A. Sontag April 9,

4 Math HWK b Solns contd 8. p99 Problem, 8. p99. Let T: R R be the linear transfmation defined by T(x, x, x ) = (x x, x x, x x ) (a) Find the matrix f T relative to the basis B = {v, v, v }, given that v = (,,), v = (,, ), v = (,,) (b) Verify that this matrix functions as advertised. In other wds, verify that multiply the B- codinate matrix f v on the left by this matrix always produces the B-codinate matrix f T(v). Solution. (a). T(v ) = (,,) = v v T(v ) = (,, ) = v + v + v T(v ) = (,, ) = v + v v so the required matrix is A = (b). Given v R, with v = av + bv + cv we have T(v) = T (a(,, ) + b(,,) + c(,,)) = T(a + c, b + c, a + b) = (a b, b a, c b) Meover, Av B = a b + c a b = a + b + c c b c so we should have (a b + c)v + ( a + b + c)v + ( b c)v = (a b, b a, c b) Page of 9 A. Sontag April 9,

5 Math HWK b Solns contd 8. p99 (a b + c)(,, ) +( a + b + c)(,,) +( b c)(,, ) = (a b, b a, c b) (a b + c,, a b + c) +(, a + b + c, a + b + c) +( b c, b c, ) = (a b, b a, c b) (a b + c + + b c, a + b + c + b c, a b + c a + b + c + ) = (a b, b a, c b) which does hold. (Vertical notation would have been a better choice here, but I won t retype.) Problem 9, 8. p99. Let and let v =, v = A = 5 be the matrix f the linear transfmation T : R R relative to the basis B = {v, v }. (a) Find T(v ) B and T(v ) B. (b) Find T(v ) and T(v ). (c) Find a fmula f T( x x ). Page 5 of 9 A. Sontag April 9,

6 Math HWK b Solns contd 8. p99 (d) Use the fmula from (c) to compute T( ). Solution. (a). T(v ) B = first column of A = T(v ) B = second column of A = 5 (b). From (a), we have (c) We need to write x x and solve f a, b you should get T(v ) = v v = T(v ) = v + 5v = + = = 9 as a linear combination of v and v. If you set up the system a + b = x x a = 7 (x + x ), b = 7 ( x + x ) Therefe T ( x x ) = T( 7 (x + x )v + 7 ( x + x )v ) = 7 (x + x )T(v ) + 7 ( x + x )v )T(v ) = 7 (x + x ) ( x + x )v ) 9 = 7 (8x + x ) 7 ( 7x + x ) = 8 x 7 7 x (d). T( ) = = Page of 9 A. Sontag April 9,

7 Math HWK b Solns contd 8. p99 Problem, 8. p99. Let A = 5 be the matrix f a linear transfmation T : P P relative to the basis B = {v, v, v } where v, v, v are given by v (x) = x + x, v (x) = + x + x, v (x) = + 7x + x (a) Find T(v ) B, T(v ) B, T(v ) B. (b) Find T(v ), T(v ), T(v ). (c) Find a fmula f T(a + a x + a x ). (d) Use the fmula from (c) to find T( + x ). Solution. (a). T(v ) B = first column of A = T(v ) B = second column of A = T(v ) B = third column of A = 5 (b) Use (a) to find: T(v ) = v + v + v = (x + x ) + ( + x + x ) + ( + 7x + x ) = + 5x + 9x T(v ) = v + v v = (x + x ) ( + 7x + x ) = 5x + 5x T(v ) = v + 5v + v = (x + x ) + 5( + x + x ) + ( + 7x + x ) = 7 + x + 5x (c). Write a + a x + a x = a(x +x )+ b( +x +x )+ c(+ 7x+ x ). Equate coefficients to find b + c = a a + b + 7c = a a + b + c = a Page 7 of 9 A. Sontag April 9,

8 Math HWK b Solns contd 8. p99 Now solve f a, b, and c. (This might be a good time to use Joy of Mathematica.) You should get a = (a a + a ) b = 8 ( 5a + a a ) c = 8 (a + a a ) Therefe T(a + a x + a x ) = T(av + bv + cv ) = (a a + a )( + 5x + 9x ) + 8 ( 5a + a a )( 5x + 5x ) + 8 (a + a a )(7 + x + 5x ) = (9a a + 89a ) + 8 (a a + 7a )x + (a a + 7a )x With Joy of Mathematica, this last simplification isn t quite so tedious. (Use the Simplify command under the Algebra menu.) (d) T( + x ) = (9 + 89) + ( + 7)x + 8 ( + 7)x = + 5x + x Problem 5, 8. p99. Show that if T : V V is a contraction dilation of V, then the matrix f T relative to any basis f V is a positive scalar multiple of the identity matrix. Solution. F this problem to make sense we must assume that V is finite-dimensional. Say V has dimension n. Assume, as indicated, that T : V V is a contraction dilation by the fact k (f some k > ). This means that T(v) = kv f every v. Let B = {v,..., v n } be a basis f V, and let A be the matrix that represents T relative to the basis B. Then we must have T(v ) = kv, T(v ) = kv, and so fth f all the n basis vects. This tells us that the first column of A has k at the top and all its other entries. The second column of A has k in the second position, and all its other entries. Me generally, the j-th column of A will have k in the j-th position and all its other entries. But this description of A tells us that A = ki, where I is the n n identity matrix. Note, in particular, that with k = we have proven that the identity transfmation has the identity matrix as its representing matrix, no matter what basis is used f V. Page 8 of 9 A. Sontag April 9,

9 Math HWK b Solns contd 8. p99 Problem, 8. p99. Assume that B = {v, v, v, v } is a basis f some vect space V. Find the matrix with respect to B f the linear operat T : V V defined by the conditions T(v ) = v, T(v ) = v, T(v ) = v, T(v ) = v Solution. Finally, a nice simple problem. Denote the desired matrix by A. From T(v ) = v we conclude that the first column of A has the entries,,,, in that der. The entries in the second column are,,,, in that der. Those in the third column are,,,, and those in the fourth are,,,. Thus A = Problem 8b, 8. p99. Let D : P P be the differentiation operat (i.e. D(p) is the derivative function p ). Find the matrix f D relative to the basis B = {p, p, p }, given that p (x) =, p (x) = x, p (x) = x + 8x Solution. Let A be the specified matrix f D. A little bit of calculus mixed with a little bit of algebra gives us D(p ) = = p + p + p D(p ) = = p + p + p D(p ) = + x = p p + p so that A = = 9. Page 9 of 9 A. Sontag April 9,