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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

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
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Hello, please I need a step by step and a well self-explanatory solution to questions (a.1), (a.2) and (a.3). I don't have a strong background in Algebra. HELP ME

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
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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