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 . 29 . Prove that (−1) u = − u. [ Hint : Show that u + (−1) u = 0 . Use some axioms and the results of Exercises 26 and 27.]
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 . 29 . Prove that (−1) u = − u. [ Hint : Show that u + (−1) u = 0 . Use some axioms and the results of Exercises 26 and 27.]
Solution Summary: The author explains how the result (-1)u=-u is proved.
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.
29. Prove that (−1)u = −u. [Hint: Show that u + (−1)u = 0. Use some axioms and the results of Exercises 26 and 27.]
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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