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 . 26 . Complete the following proof that − u is the unique vector in V such that u + (− u ) = 0 . Suppose that w satisfies u + w = 0 . Adding − u to both sides, we have ( − u ) + [ u + w ] = ( − u ) + 0 [ ( − u ) + u ] + w = ( − u ) + 0 b y A x i o m _ _ _ _ _ _ ( a ) 0 + w = ( − u ) + 0 b y A x i o m _ _ _ _ _ _ ( b ) w = − u b y A x i o m _ _ _ _ _ _ ( c )
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 . 26 . Complete the following proof that − u is the unique vector in V such that u + (− u ) = 0 . Suppose that w satisfies u + w = 0 . Adding − u to both sides, we have ( − u ) + [ u + w ] = ( − u ) + 0 [ ( − u ) + u ] + w = ( − u ) + 0 b y A x i o m _ _ _ _ _ _ ( a ) 0 + w = ( − u ) + 0 b y A x i o m _ _ _ _ _ _ ( b ) w = − u b y A x i o m _ _ _ _ _ _ ( c )
Solution Summary: The author explains the proof that -u is a unique vector in the vector space V.
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.
26. Complete the following proof that −u is the unique vector in V such that u + (−u) = 0. Suppose that w satisfies u + w = 0. Adding −u to both sides, we have
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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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