Set V = {(x, y) | x ER, YER}. Define addition of vectors by the rule (x, y) + (x, y) = (y + y',x+x') and define scalar multiplication by the rule (x, y) c(x, y) = (y,x) (0,0) if c> 0 if c < 0 if c = 0 For example, (1,2)+(3,4)= (6,4) and -2(3, 4) =(4,3). For each of the vector space axioms (A1)-(A4), (M1)-(M4), either prove that the axiom holds or prove that it does not hold. Use the axioms as described in the lecture notes. Here are a few things to keep in mind. • To disprove (A2), you would need to show that no vector z = (a, b) has the property that for all u = (x, y), u + z = u and z+u = u. • If (A2) is false, (A3) is automatically false as it relies on the existence of a zero vector. • To disprove (A3), you would need to give a specific example of a vector u such that for all other vectors w at least one of u + w = z or w+u = z is false.
Set V = {(x, y) | x ER, YER}. Define addition of vectors by the rule (x, y) + (x, y) = (y + y',x+x') and define scalar multiplication by the rule (x, y) c(x, y) = (y,x) (0,0) if c> 0 if c < 0 if c = 0 For example, (1,2)+(3,4)= (6,4) and -2(3, 4) =(4,3). For each of the vector space axioms (A1)-(A4), (M1)-(M4), either prove that the axiom holds or prove that it does not hold. Use the axioms as described in the lecture notes. Here are a few things to keep in mind. • To disprove (A2), you would need to show that no vector z = (a, b) has the property that for all u = (x, y), u + z = u and z+u = u. • If (A2) is false, (A3) is automatically false as it relies on the existence of a zero vector. • To disprove (A3), you would need to give a specific example of a vector u such that for all other vectors w at least one of u + w = z or w+u = z is false.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.3: Lines And Planes
Problem 18EQ
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Transcribed Image Text:Set V = {(x, y) | xER, YER}. Define addition of vectors by the rule
(x, y) + (x, y) = (y + y',x+x')
and define scalar multiplication by the rule
(x,y)
c(x, y) = (y,x)
(0,0)
if c>0
if c < 0
if c = 0
For example, (1, 2) + (3, 4) = (6,4) and -2(3, 4) =(4,3).
For each of the vector space axioms (A1)-(A4), (M1)-(M4), either prove that the axiom holds or prove that it does
not hold. Use the axioms as described in the lecture notes. Here are a few things to keep in mind.
• To disprove (A2), you would need to show that no vector z = (a, b) has the property that for all u = (x, y),
u + z = u and z+u = u.
• If (A2) is false, (A3) is automatically false as it relies on the existence of a zero vector.
• To disprove (A3), you would need to give a specific example of a vector u such that for all other vectors wat
least one of u + w = z or w+u = z is false.
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