Homework Answers
Let a T: M2x2(R) + P2(R), 6 d H (2a +b)x2 + (6 – c)x +(c...
Let a T: M2x2(R) + P2(R), 6 d H (2a +b)x2 + (6 – c)x +(c – 3d). с Let B = 9 (6 8), (8 5), (1 3), ( )) (CO 11),( ( 1),66 1 B' 1 1 ? :-)) C = (x²,2,1) C' = (x + 2,2 +3,22 – 2x – 6). Is T invertible? (1pt) O Yes O No
Q1 17 Points Let T: M2x2(R) P2(R), H (2a +b)x2 + (6 – c)x +(c –...
Q1 17 Points Let T: M2x2(R) P2(R), H (2a +b)x2 + (6 – c)x +(c – 3d). Let B = (16 0) (0 :), (1 o) 9)) = (6 7')(*: -) ) 6 :-)) B' = C = (x2,æ, 1) C'= (x + 2, x + 3, x2 – 2x – 6). You may assume that all of the above are bases for the corresponding vector spaces. Q1.1 2 Points Show that T is linear. Q1.2 9 Points Compute [T),...
Show that T is linear Q1 17 Points Let b T: M2x2(R) + P2 (R), H...
Show that T is linear
Q1 17 Points Let b T: M2x2(R) + P2 (R), H (2a+b)x2 + (b – c)x+(c – 3d). с d Let 1 0 0 0 B = (( b); C8 1 0 0 0 0 1 :)C. 11), 1) (7.1)) i), (6 ;)) 1 0 1 B = (CO 2 -1 1 1 1 C = (x², x, 1) C' = (x + 2, x + 3, x2 – 2x – You may assume that...
(11) Let the linear transformation T : M2x2(R) + P2 (R) be defined by T (+...
(11) Let the linear transformation T : M2x2(R) + P2 (R) be defined by T (+ 4) = a +d+(6–c)n +(a–b+c+d)a? (1-1) (i) (3 marks) Find a basis for the T-cyclic subspace generated by (ii) (3 marks) Determine rank(T).
Let T. M2(R) →P2(R) be defined by T.(Iga)-(+b) + (b+c) Let T2: P2 (R) → Pl...
Let T. M2(R) →P2(R) be defined by T.(Iga)-(+b) + (b+c) Let T2: P2 (R) → Pl (R) be defined by Tap(x))-p' (x) (c+ d)x2 2. Find Ker(T2 . T) and find a basis for Ker(T2。T).
6.Let W={(a +b-c,2a +3b, -a +3c,-b-2c): a,b, CER) a) For what value of n is W...
6.Let W={(a +b-c,2a +3b, -a +3c,-b-2c): a,b, CER) a) For what value of n is W isomorphic to R"?, clear answer the question and justify your answer b) Find an isomorphism T:R" W for the value of n you found in part (a).Please make it clear from your work that your function T is really an isomorphism. 7.Let T:P2(R) – P2(R) be a linear transformation such that T(x2)=x2 a) Prove that if there exist distinct linearly independent polynomials 2,9 €...
(9 marks) Consider L: M2x2(R) + P2 (R) defined by a L b d = a...
(9 marks) Consider L: M2x2(R) + P2 (R) defined by a L b d = a + (b + c)2 + dx?. (a) Show that L is a linear transformation, that is, show that L(sA+B) = 8L(A) + tL(B) for any A, B E M2x2(R) and any st ER. (b) Consider p(x) = ao+a1x + a2x2 € P2(R). Show that L is onto by showing that L(A) = P(x) for some matrix A € M2x2(R). Note that you must give...
Let V P2(R) and let T V-V be a linear transformation defined by T(p)-q, where (x)(r...
Let V P2(R) and let T V-V be a linear transformation defined by T(p)-q, where (x)(r p (r Let B = {x, 1 + x2, 2x-1} be a basis of V. Compute [TIB,B, and deduce if it is eigenvectors basis of
1 #6: Let T: P2 → p2 be defined by T(ao +ajx + a2 x2) =...
1 #6: Let T: P2 → p2 be defined by T(ao +ajx + a2 x2) = (Tao + 381 +8a2) – (a1 + 36a2)x+ 20 x2 Find the eigenvalues of T. Enter any repeated eigenvalues as often as they repeat. em #6:
Let T:P1→P2T:P1→P2 be a linear transformation defined by T(a+bx)=3a−2bx+(a+b)x2.T(a+bx)=3a−2bx+(a+b)x2. (a) Find range(T)range(T) and give a basis...
Let T:P1→P2T:P1→P2 be a linear transformation defined by T(a+bx)=3a−2bx+(a+b)x2.T(a+bx)=3a−2bx+(a+b)x2. (a) Find range(T)range(T) and give a basis for range(T)range(T). (b) Find ker(T)ker(T) and give a basis for ker(T)ker(T). (c) By justifying your answer determine whether TT is onto. (d) By justifying your answer determine whether TT is one-to-one. (e) Find [T(7+x)]B[T(7+x)]B, where B={−1,−2x,4x2}B={−1,−2x,4x2}.
Let a T: M2x2(R) + P2(R), 6 d H (2a +b)x2 + (6 – c)x +(c – 3d). с Let B = 9 (6 8), (8 5), (1 3), ( )) (CO 11),( ( 1),66 1 B' 1 1 ? :-)) C = (x²,2,1) C' = (x + 2,2 +3,22 – 2x – 6). Is T invertible? (1pt) O Yes O No
Q1 17 Points Let T: M2x2(R) P2(R), H (2a +b)x2 + (6 – c)x +(c – 3d). Let B = (16 0) (0 :), (1 o) 9)) = (6 7')(*: -) ) 6 :-)) B' = C = (x2,æ, 1) C'= (x + 2, x + 3, x2 – 2x – 6). You may assume that all of the above are bases for the corresponding vector spaces. Q1.1 2 Points Show that T is linear. Q1.2 9 Points Compute [T),...
Show that T is linear
Q1 17 Points Let b T: M2x2(R) + P2 (R), H (2a+b)x2 + (b – c)x+(c – 3d). с d Let 1 0 0 0 B = (( b); C8 1 0 0 0 0 1 :)C. 11), 1) (7.1)) i), (6 ;)) 1 0 1 B = (CO 2 -1 1 1 1 C = (x², x, 1) C' = (x + 2, x + 3, x2 – 2x – You may assume that...
(11) Let the linear transformation T : M2x2(R) + P2 (R) be defined by T (+ 4) = a +d+(6–c)n +(a–b+c+d)a? (1-1) (i) (3 marks) Find a basis for the T-cyclic subspace generated by (ii) (3 marks) Determine rank(T).
Let T. M2(R) →P2(R) be defined by T.(Iga)-(+b) + (b+c) Let T2: P2 (R) → Pl (R) be defined by Tap(x))-p' (x) (c+ d)x2 2. Find Ker(T2 . T) and find a basis for Ker(T2。T).
6.Let W={(a +b-c,2a +3b, -a +3c,-b-2c): a,b, CER) a) For what value of n is W isomorphic to R"?, clear answer the question and justify your answer b) Find an isomorphism T:R" W for the value of n you found in part (a).Please make it clear from your work that your function T is really an isomorphism. 7.Let T:P2(R) – P2(R) be a linear transformation such that T(x2)=x2 a) Prove that if there exist distinct linearly independent polynomials 2,9 €...
(9 marks) Consider L: M2x2(R) + P2 (R) defined by a L b d = a + (b + c)2 + dx?. (a) Show that L is a linear transformation, that is, show that L(sA+B) = 8L(A) + tL(B) for any A, B E M2x2(R) and any st ER. (b) Consider p(x) = ao+a1x + a2x2 € P2(R). Show that L is onto by showing that L(A) = P(x) for some matrix A € M2x2(R). Note that you must give...
Let V P2(R) and let T V-V be a linear transformation defined by T(p)-q, where (x)(r p (r Let B = {x, 1 + x2, 2x-1} be a basis of V. Compute [TIB,B, and deduce if it is eigenvectors basis of
1 #6: Let T: P2 → p2 be defined by T(ao +ajx + a2 x2) = (Tao + 381 +8a2) – (a1 + 36a2)x+ 20 x2 Find the eigenvalues of T. Enter any repeated eigenvalues as often as they repeat. em #6:
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