Exercises 25–29 show how the axioms for a vector space V can be used to prove the elementary properties described after the definition of a vector space. Fill in the blanks with the appropriate axiom numbers. Because of Axiom 2, Axioms 4 and 5 imply, respectively, that 0 + u = u and − u + u = 0 for all u . 27 . Fill in the missing axiom numbers in the following proof that 0 u = 0 for every u in V . 0 u = (0 + 0) u = 0 u + 0 u by Axiom _____ (a) Add the negative of 0 u to both sides: 0 u + [ − 0 u ] = [ 0 u + 0 u ] + ( − 0 u ) 0 u + [ − 0 u ] = 0 u + [ 0 u + ( − 0 u ) ] b y A x i o m _ _ _ _ _ _ ( b ) 0 = 0 u + 0 b y A x i o m _ _ _ _ _ _ ( c ) 0 = 0 u b y A x i o m _ _ _ _ _ _ ( d )
Exercises 25–29 show how the axioms for a vector space V can be used to prove the elementary properties described after the definition of a vector space. Fill in the blanks with the appropriate axiom numbers. Because of Axiom 2, Axioms 4 and 5 imply, respectively, that 0 + u = u and − u + u = 0 for all u . 27 . Fill in the missing axiom numbers in the following proof that 0 u = 0 for every u in V . 0 u = (0 + 0) u = 0 u + 0 u by Axiom _____ (a) Add the negative of 0 u to both sides: 0 u + [ − 0 u ] = [ 0 u + 0 u ] + ( − 0 u ) 0 u + [ − 0 u ] = 0 u + [ 0 u + ( − 0 u ) ] b y A x i o m _ _ _ _ _ _ ( b ) 0 = 0 u + 0 b y A x i o m _ _ _ _ _ _ ( c ) 0 = 0 u b y A x i o m _ _ _ _ _ _ ( d )
Solution Summary: The author explains that the proof has been completed using Axioms a. 8, b. 3, c. 5, and d.
Exercises 25–29 show how the axioms for a vector space V can be used to prove the elementary properties described after the definition of a vector space. Fill in the blanks with the appropriate axiom numbers. Because of Axiom 2, Axioms 4 and 5 imply, respectively, that 0 + u = u and −u + u = 0 for all u.
27. Fill in the missing axiom numbers in the following proof that
0u = 0 for every u in V.
0u = (0 + 0)u = 0u + 0u by Axiom _____ (a)
Add the negative of 0u to both sides:
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Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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