MATH 110 Spring 2015 Homework 6 Solutions


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1 MATH 110 Spring 2015 Homework 6 Solutions Section Let α denote the standard basis for V = R 3. Let α = {e 1, e 2, e 3 } denote the dual basis of α for V. We would first like to show that β = {f 1, f 2, f 3 } is a basis of V. f 1 = e 1 2e 2, f 2 = e 1 + e 2 + e 3, f 3 = e 2 3e 3. Suppose af 1 + bf 2 + cf 3 = 0 = a(e 1 2e 2 ) + b(e 1 + e 2 + e 3 ) + c(e 2 3e 3 ) = (a + b)e 1 + ( 2a + b + c)e 2 + (b 3c)e 3 (1) Since α = {e 1, e 2, e 3 } is a basis of V, a + b = 2a + b + c = b 3c = 0. Solving this we get a = b = c = 0. This shows that {f 1, f 2, f 3 } is a linearly independent subset in V. Since dim(v ) = 3, β = {f 1, f 2, f 3 } is a basis. Now we need to find a basis β = {g 1, g 2, g 3 } of V = R 3 such that β is the dual basis of β. (Method 1) From the definition of the dual basis, we know that (x, y, z) = (x 2y)g 1 +(x+y+z)g 2 +(y 3z)g 3 Let G denote the matrix with column vectors being g 1, g 2, g 3, then the equation above is equivalent to the matrix equation below, G A = I, where A is the matrix Then G = A 1, and is given by The elements of the basis β are the columns of the matrix G. More explicitly, β = {(0.4, 0.3, 0.1), (0.6, 0.3, 0.1), (0.2, 0.1, 0.3)}. 1
2 (Method 2) We discuss a different method below using Theorem 2.25 of the textbook. The change of coordinate matrix from β to α, [I] α β (Here I denotes the identity transformation of V ), is given by Note that the identity transformation on V is the transpose of the identity transformation on V, according to the definition in Theorem 2.25 of the textbook. Then by Theorem 2.25, the change of coordinate matrix from β to α is the transpose of the change of coordinate matrix from α to β [I] β α. Therefore, [I] β α is given by ([I] α β )t = [I] β α Then the change of coordinate matrix from β to α, [I] α β = ([I]β α) 1 is obtained by calculating the inverse of the matrix above, and is given by The elements of the basis β are the columns of the matrix [I] α β. More explicitly, (a) T t (f) (R 2 ) is given by β = {(0.4, 0.3, 0.1), (0.6, 0.3, 0.1), (0.2, 0.1, 0.3)}. T t (f)(x, y) = ft (x, y) = f(3x + 2y, x) = 2(3x + 2y) + x = 7x + 4y. (b) Let [T t ] β = ([T t (f 1 )] β, [T t (f 2 )] β ) be ( a b c d such that T t (f 1 ) = af 1 + cf 2 and T t (f 2 ) = bf 1 + df 2. Let β = {e 1 = (1, 0), e 2 = (0, 1)} denote the standard ordered basis of R 2. T t (f 1 )(1, 0) = f 1 T (1, 0) = f 1 (3, 1) = 3 = (af 1 + cf 2 )(1, 0) = a. ) 2
3 T t (f 1 )(0, 1) = f 1 T (0, 1) = f 1 (2, 0) = 2 = (af 1 + cf 2 )(0, 1) = c. T t (f 2 )(1, 0) = f 2 T (1, 0) = f 2 (3, 1) = 1 = (bf 1 + df 2 )(1, 0) = b. T t (f 2 )(0, 1) = f 2 T (0, 1) = f 2 (2, 0) = 0 = (bf 1 + df 2 )(0, 1) = d. Therefore, [T t ] β = ( ) (c) [T ] β = ([T (1, 0)] β, [T (0, 1)] β ) = Then ([T ] β ) t = ( ( This is the same as [T t ] β as predicted by Theorem (a) We can check that f i (p(x) + cq(x)) = p(c i ) + cq(c i ) = f i (p(x)) + cf i (q(x)), so f i is linear and is in V?. Also we know that dim(v? ) = dim(v ) = dim(p n (F )) = n + 1. So now it?s enough to show that {f 0, f 1,..., f n } is linearly lindependent. Assume that Σ n i=0 a i f i = 0 for some scalars a i. We may define polynomials p i (x) = Π j i (x?cj) such that we know p i (c i ) 0 but p i (c j ) = 0 for all j i. Now fix an i {0, 1,..., n}? n k=0 a kf k (p i ) = a i f i (p i ) = a i p i (c i ) = 0. Since p i (c i ) 0, a i = 0. Therefore a i = 0 for all i and {f 0, f 1,..., f n } is a basis. (b) By the Corollary after Theorem 2.26 we have an ordered basis? = p 0, p 1,..., p n for V such that f 1, f 2,..., f n defined in (a) is its dual basis. Then we know that f j (p i ) = p i (c j ) = δ ij. Since β is a basis, every polynomial in V can be written as a unique linear combination of elements of β. Fix an integer b {0,..., n} Suppose a polynomial q has the property q(c j ) = δ bj, then we could write q as a linear combination q = Σ k a k p k. Plug in c j, j = 0,..., n, we get a b = 1 and a j = 0 for all j b. Therefore, q = p b. Since b is arbitrary, we know that {p 0,..., p n } is unique. (c) First, q(x) = Σ n i=0 a ip i (x) has the property q(c i ) = a i for 0 i n. Now suppose there is another polynomial r(x) that has this property. Express r(x) as a linear combination of the elements of the basis β defined in (b), r(x) = Σ n i=0 b ip i (x). Plug in c j, j = 0,..., n we get a i = b i for all 0 i n. Therefore r(x) = q(x) = Σ n i=0 a ip i (x). (d) By previous parts of this exercise, every polynomial p(x) could be uniquely written as a linear combination Σ n i=0 z ip i (x) such that z i = p(c i ). Therefore we have ) ) p(x) = Σ n i=0p(c i )p i (x). (e) Since there are only finitely many terms in the summation above, the order of integration and summation can be exchanged. Therefore we get 3
4 b a p(t)dt = Σ n i=0p(c i ) b a p i (t)dt. (a)we can check that if f, g S 0, c F, then f + g and cf are elements in S 0. This is because (f + g)(x) = f(x) + g(x) = 0 and (cf)(x) = cf(x) = 0. The zero functional is an element in S 0. (b) Let v 1, v 2,..., v k be the basis of W. Since x / W, we know that {v 1, v 2,..., v k+1 = x} is a linearly independent set and hence we can extend it to a basis {v 1, v 2,..., v n } for V. So we can define a linear functional f W 0 such that f(v i ) = δ i,(k+1) for i = 1,, k, k + 1,, n. Thus f is the desired functional. (c) Let W be the subspace span(s). We first prove that W 0 = S 0. Since every functional that is zero at every w W must be a functional that is zero at every s S, we know W 0 S 0. On the other hand, if a linear functional f has the property that f(x) = 0 for all xins, we can deduce that f(y) = 0 for all y W = span(s). Hence, we know that S 0 W 0 and therefore W 0 = S 0. Since (W 0 ) 0 = (S 0 ) 0 and span(ψ(s)) = ψ(w ) by the fact ψ is an isomorphism, we can just prove that (W 0 ) 0 = ψ(w ). Next, by Theorem 2.26 we may assume every element in (W 0 ) 0?V?? has the form ˆx for some x V. Let ˆx be an element in (W 0 ) 0. We have that ˆx(f) = f(x) = 0 if f W 0. Now if x is not an element of W, by part b, there exists some functional f?w 0 such that f(x) 0. This is a contradiction. So we know that ˆx is an element in ψ(w ) and (W 0 ) 0 ψ(w ). For the converse, we may assume that ˆx is an element in ψ(w ). Then for all f W 0 we have that ˆx(f) = f(x) = 0 since x is an element in W. So we know that ψ(w ) (W 0 ) 0. Therefore (W 0 ) 0 = ψ(w ) and we get the desired conclusion. (d) If W 1 = W 2, then we have W 0 1 = W 0 2. For the converse, if W 0 1 = W 0 2, then we have ψ(w 1 ) = (W 0 1 ) 0 = (W 0 2 ) 0 = ψ(w 2 ). Since that ψ is an isomorphism, W 1 = ψ?1 (ψ?(w 1 )) = ψ?1 (ψ(w 2 )) = W 2. (e) If f is an element in (W 1 + W 2 ) 0, we have that f(w 1 + w 2 ) = 0 for all w 1 W 1 and w 2 W 2. So we know that f(w 1 + 0) = 0 and f(0 + w 2 ) = 0 for all w 1 W 1 and w 2 W 2. This means f is an element in W 0 1 W 0 2. For the converse, if f is an element of W 1 0 W 2 0, we have that f(w 1 + w 2 ) = f(w 1 ) + f(w 2 ) = 0 for all w 1 W 1 and w 2 W 2. Hence we have that f is an element in (W 1 + W 2 ) If T t (f) = ft = 0, then f(y) = 0 for all y R(T ) and hence f (R(T )) 0. If f (R(T )) 0, then f(y) = 0 for all y R(T ) and hence T t (f)(x) = f(t (x)) = 0 for all x V. This means f is an element in N(T t ). 4
5 Section (a) False. The system 0x = 1 has no solution. (b) False. The system that 0x = 0 has infinitely many solutions. (c) True. It at least has the zero solution. (d) False. The system that 0 (x 1, x 2,, x n ) = 0 has infinitely many solutions. (e) False. The system that 0 (x 1, x 2,, x n ) = a, a being a nonzero vector in R n, has no solution. (f) False. The system 0x = 1 has no solution but the homogeneous system corresponding to it has infinitely many solutions. (g) True. If Ax = 0 and A is an invertible n n matrix, then we know x = A 1 0 = 0. (h) False. The system x = 1 has solution set {1}, which is not a subspace of F (b,c) (b) Subtract the first equation from the second equation and we get 3x 1 = x 3. Then from the first equation we get x 2 = x 3 x 1 = 2x 1. Therefore all solutions have the form {x 1 (1, 2, 3) x 1 R}. The set {(1, 2, 3)} is a basis of the solution set and the dimension is 1. (c) Add the two equations we get 3x 1 + 3x 2 = 0, therefore x 2 = x 1. From the first equation we find x 3 = x 1 + 2x 2 = x 1 2x 1 = x 1. Therefore all solutions have the form {x 1 (1, 1, 1) x 1 R} The set {(1, 1, 1)} is a basis of the solution set and the dimension is (b,c) Pick one particular solution b to this nonhomogeneous system of linear equations. The all solutions to this nonhomogeneous system is given by the sum of b and one solution to the corresponding homogeneous system of linear equations. (b) ( 2 3, 1 3, 0) is one solution to this nonhomogeneous system. Then all solutions have the form {( 2 3, 1 3, 0) + c (1, 2, 3) c R}. 5 (c) (3, 0, 0) is one particular solution to this nonhomegeneous system. Then all solutions have the form {(3, 0, 0) + c (1, 1, 1) c R}. Let A be the n n zero matrix. The system Ax = 0 has infinitely many solutions. 5
6 8(a) Solve a + b = 1, b 2c = 3, a + 2c = 2, and we get a = 0, b = 1, c = 1 is a solution. Therefore v R(T ). 6
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