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.
Compute the products in Exercises 1–4 using (a) the definition, as in Example 1, and (b) the row–vector rules for computer Ax, or, the rule for computing a product Ax in which the i th entry of Ax is the sum of the products of corresponding entries from row i of A and from the vector x. If a product is undefined, explain why.
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