ZOOM Let V and W be vector spaces over a field K, and f: V -> W a linear map. (a) Let f R4 -+ R4 be a linear map defined by f(e) = ie; for 1 < i < 4 where (e1,.., e4) I, the basis consisting of unit vectors. Let 1 1 1 1 Н — 1 -1 1 E R4x4 1 -1 1 (a.1) Compute H'H and H. Deduce that the columns of H form a basis for R4 (a.2) Find the dual basis of H. (Describe the linear forms explicitly.) (a.3 Find the matrix of f under the basis H of R4 (b) Suppose V and W have the same fiite dimension n. Prove the following: (b.1) If f is injective, then for any basis u,..., u of V, the images f(u1),... ,f(1,) form a basis for W, hencef is onto; (b.2) If f is onto, then for any basis u, . .. , u Oof V, the images f(u1), ... ,f(u,) form a basis for W, hence f is injective Remark. Note that neither of above is true when the dimension is infinite (see examples in class). In your proof, you may use the fact that any set of independent vectors in a vector space can be extended to basis and any set of dependent vectors contains a subset that is a basis. (c) Let X C V be a linear subspace such that X C Nullf), and let T: V -> V/X be the natural surjection T(v) = 7. (c.1) Let f V/X > W be given by f(7) defined, that is, v, vi E V are such that u = v, (mod X), then (v) = T(v1), so f(7) is independent of the choice of representative of .) Prove that f is a linear transformation such that foT f and f(V/X) = f(V) (c.2) Prove that if S: V/X - W is a a linear transformation such that S oT f f(v). (First shows that this is well- then S
ZOOM Let V and W be vector spaces over a field K, and f: V -> W a linear map. (a) Let f R4 -+ R4 be a linear map defined by f(e) = ie; for 1 < i < 4 where (e1,.., e4) I, the basis consisting of unit vectors. Let 1 1 1 1 Н — 1 -1 1 E R4x4 1 -1 1 (a.1) Compute H'H and H. Deduce that the columns of H form a basis for R4 (a.2) Find the dual basis of H. (Describe the linear forms explicitly.) (a.3 Find the matrix of f under the basis H of R4 (b) Suppose V and W have the same fiite dimension n. Prove the following: (b.1) If f is injective, then for any basis u,..., u of V, the images f(u1),... ,f(1,) form a basis for W, hencef is onto; (b.2) If f is onto, then for any basis u, . .. , u Oof V, the images f(u1), ... ,f(u,) form a basis for W, hence f is injective Remark. Note that neither of above is true when the dimension is infinite (see examples in class). In your proof, you may use the fact that any set of independent vectors in a vector space can be extended to basis and any set of dependent vectors contains a subset that is a basis. (c) Let X C V be a linear subspace such that X C Nullf), and let T: V -> V/X be the natural surjection T(v) = 7. (c.1) Let f V/X > W be given by f(7) defined, that is, v, vi E V are such that u = v, (mod X), then (v) = T(v1), so f(7) is independent of the choice of representative of .) Prove that f is a linear transformation such that foT f and f(V/X) = f(V) (c.2) Prove that if S: V/X - W is a a linear transformation such that S oT f f(v). (First shows that this is well- then S
Advanced Engineering Mathematics
10th Edition
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please help with the solutions to questions (b.1), (b,2) and (c.1.) Please, give a much detailed and self-explanatory solution and I will be glad to you. thank you
Transcribed Image Text:ZOOM
Let V and W be vector spaces over a field K, and f: V -> W a linear map.
(a) Let f R4 -+ R4 be a linear map defined by f(e) = ie; for 1 < i < 4 where
(e1,.., e4) I, the basis consisting of unit vectors. Let
1
1
1
1
Н —
1
-1
1
E R4x4
1
-1
1
(a.1) Compute H'H and H. Deduce that the columns of H form a basis for R4
(a.2) Find the dual basis of H. (Describe the linear forms explicitly.)
(a.3 Find the matrix of f under the basis H of R4
(b) Suppose V and W have the same fiite dimension n. Prove the following:
(b.1) If f is injective, then for any basis u,..., u of V, the images f(u1),... ,f(1,)
form a basis for W, hencef is onto;
(b.2) If f is onto, then for any basis u, . .. , u Oof V, the images f(u1), ... ,f(u,) form
a basis for W, hence f is injective
Remark. Note that neither of above is true when the dimension is infinite (see
examples in class). In your proof, you may use the fact that any set of independent
vectors in a vector space can be extended to basis and any set of dependent vectors
contains a subset that is a basis.
(c) Let X C V be a linear subspace such that X C Nullf), and let T: V -> V/X be
the natural surjection T(v) = 7.
(c.1) Let f V/X > W be given by f(7)
defined, that is, v, vi E V are such that u = v, (mod X), then (v) = T(v1), so
f(7) is independent of the choice of representative of .) Prove that f is a linear
transformation such that foT f and f(V/X) = f(V)
(c.2) Prove that if S: V/X - W is a a linear transformation such that S oT f
f(v). (First shows that this is well-
then S
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