Please explain the 8 rules of addition and scalar multiplication for a vector space and provide solutions in detail. (a) Suppose addition in R? adds an extra 1 to each component, so that (3, 1) + (5,0) = (9,2) in-stead of (8,1). With scalar multiplication unchanged, which of the 8 rules of addition and scalarmultiplication for a vector space are broken?(b) Show that the set of all positive real numbers, with x+y and cx redefined to equal the usual xyand x, is a vector space, or indicate why it is not. What is the "zero vector"?

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Answered: (a) Suppose addition in R? adds an…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.1: Vector Spaces And Subspaces

Problem 2EQ

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Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Illustrate properties 110 of Theorem 4.2 for u=(2,1,3,6), v=(1,4,0,1), w=(3,0,2,0), c=5, and d=2. THEOREM 4.2Properties of Vector Addition and Scalar Multiplication in Rn. Let u,v, and w be vectors in Rn, and let c and d be scalars. 1. u+v is vector in Rn. Closure under addition 2. u+v=v+u Commutative property of addition 3. (u+v)+w=u+(v+w) Associative property of addition 4. u+0=u Additive identity property 5. u+(u)=0 Additive inverse property 6. cu is a vector in Rn. Closure under scalar multiplication 7. c(u+v)=cu+cv Distributive property 8. (c+d)u=cu+du Distributive property 9. c(du)=(cd)u Associative property of multiplication 10. 1(u)=u Multiplicative identity property
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True or False? In Exercises 49 and 50, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a To show that a set is not a vector space, it is sufficient to show that just one axiom is not satisfied. b The set of all first-degree polynomials with the standard operations is a vector space. c The set of all pairs of real numbers of the form (0,y), with the standard operations on R2, is a vector space.

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(a) Suppose addition in R? adds an extra 1 to each component, so that (3, 1) + (5,0) = (9,2) in- stead of (8,1). With scalar multiplication unchanged, which of the 8 rules of addition and scalar multiplication for a vector space are broken?