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 . 28 . Fill in the missing axiom numbers in the following proof that c 0 = 0 for every scalar c . c 0 = c ( 0 + 0 ) = c 0 + c 0 b y A x i o m _ _ _ _ _ _ ( a ) Add the negative of c 0 to both sides: c 0 + [ - c 0 ] = [ c 0 + c 0 ] + ( − c 0 ) c 0 + [ − c 0 ] = c 0 + [ c 0 + ( − c 0 ) ] b y A x i o m _ _ _ _ _ _ ( c ) 0 = c 0 + 0 b y A x i o m _ _ _ _ _ _ ( d ) 0 = c 0 b y A x i o m _ _ _ _ _ _ ( e )
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 . 28 . Fill in the missing axiom numbers in the following proof that c 0 = 0 for every scalar c . c 0 = c ( 0 + 0 ) = c 0 + c 0 b y A x i o m _ _ _ _ _ _ ( a ) Add the negative of c 0 to both sides: c 0 + [ - c 0 ] = [ c 0 + c 0 ] + ( − c 0 ) c 0 + [ − c 0 ] = c 0 + [ c 0 + ( − c 0 ) ] b y A x i o m _ _ _ _ _ _ ( c ) 0 = c 0 + 0 b y A x i o m _ _ _ _ _ _ ( d ) 0 = c 0 b y A x i o m _ _ _ _ _ _ ( e )
Solution Summary: The author explains that the proof has been completed using Axioms a. 4, b. 7, c. 3, d. 5, and e.
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.
28. Fill in the missing axiom numbers in the following proof that c0 = 0 for every scalar c.
c
0
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Add the negative of c0 to both sides:
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[
c
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c
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−
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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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