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2. Suppose that V and W are vector spaces over F. Consider the cartesian product V x W, with vector addition and scalar multiplication defined by (V1, W1) + (v2, W2) (V1 + V2, W1 + W2) and c(v1, W1) = (cv1, cW1) for every (v1, w1), (v2, w2) E V x W and cE F. a) Show that V x W is a vector space over F. b) Suppose that || - ||v is a norm on V and || · ||w is a norm on W. Show that I|(v, w)|| = ||v||v + ||w||w defines a norm on V x W. c) Show that a sequence (vn, Wn)nEN in V xW converges to (v, w) E V xW as n oo if and only if the sequence (vn)nEN in V converges to v € V and the sequence (wn)nEN in W converges to w e W asn - 00.

2. Suppose that V and W are vector spaces over F. Consider the cartesian product V x W, with vector addition and scalar multiplication defined by (V1, W1) + (v2, W2) (V1 + V2, W1 + W2) and c(v1, W1) = (cv1, cW1) for every (v1, w1), (v2, w2) E V x W and cE F. a) Show that V x W is a vector space over F. b) Suppose that || - ||v is a norm on V and || · ||w is a norm on W. Show that I|(v, w)|| = ||v||v + ||w||w defines a norm on V x W. c) Show that a sequence (vn, Wn)nEN in V xW converges to (v, w) E V xW as n oo if and only if the sequence (vn)nEN in V converges to v € V and the sequence (wn)nEN in W converges to w e W asn - 00.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 34EQ
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2. Suppose that V and W are vector spaces over F. Consider the cartesian product V xW, with vector addition and scalar multiplication defined by (V1, W1) + (v2, w2) = (V1 + V2, W1 + w2) and c(v1, W1) = (cv1, cW1) for every (v1, w1), (v2, w2) E V x W and ce F. a) Show that V x W is a vector space over F. b) Suppose that || · ||v is a norm on V and || · ||w is a norm on W. Show that |(v, w)|| = ||v||v + ||w||w defines a norm on V x W. c) Show that a sequence (vn, wn)nEN in V x W converges to (v, w) E V x W as n→ o if and only if the sequence (vn)nEN in V converges to vE V and the sequence (wn)nEN in W converges to w e W asn → 00. d) Comment on Cauchy sequences.
Transcribed Image Text:2. Suppose that V and W are vector spaces over F. Consider the cartesian product V xW, with vector addition and scalar multiplication defined by (V1, W1) + (v2, w2) = (V1 + V2, W1 + w2) and c(v1, W1) = (cv1, cW1) for every (v1, w1), (v2, w2) E V x W and ce F. a) Show that V x W is a vector space over F. b) Suppose that || · ||v is a norm on V and || · ||w is a norm on W. Show that |(v, w)|| = ||v||v + ||w||w defines a norm on V x W. c) Show that a sequence (vn, wn)nEN in V x W converges to (v, w) E V x W as n→ o if and only if the sequence (vn)nEN in V converges to vE V and the sequence (wn)nEN in W converges to w e W asn → 00. d) Comment on Cauchy sequences.
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