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Prove that in a given vector space V, the zero vector is unique. Suppose, by way of contradiction, that there are two distinct additive identities 0 and u. Which of the following statements are then true about the vectors 0 and u,? (Select all that apply.) O The vector 0 + u, does not exist in the vector space V. O The vector 0 + u, is equal to u O The vector 0 + u, is equal to 0. O The vector 0 + u, is not equal to uo +0 O The vector 0 + u, is not equal to 0. O The vector 0 + u, is not equal to up- Which of the following is a result of the true statements that were chosen and what contradiction then occurs? O The statement u, + 0 + 0, which contradicts that u, is an additive identity. O The statement u, +0 + 0, which contradicts that u, must have an additive inverse. O The statement u, + 0+ 0 + Ug, which contradicts the commutative property. O The statement u, = 0, which contradicts that there are two distinct additive identities. O The statement u, + 0 + up. which contradicts that 0 is an additive identity. Therefore, the additive identity in a vector space is unique.

Prove that in a given vector space V, the zero vector is unique. Suppose, by way of contradiction, that there are two distinct additive identities 0 and u. Which of the following statements are then true about the vectors 0 and u,? (Select all that apply.) O The vector 0 + u, does not exist in the vector space V. O The vector 0 + u, is equal to u O The vector 0 + u, is equal to 0. O The vector 0 + u, is not equal to uo +0 O The vector 0 + u, is not equal to 0. O The vector 0 + u, is not equal to up- Which of the following is a result of the true statements that were chosen and what contradiction then occurs? O The statement u, + 0 + 0, which contradicts that u, is an additive identity. O The statement u, +0 + 0, which contradicts that u, must have an additive inverse. O The statement u, + 0+ 0 + Ug, which contradicts the commutative property. O The statement u, = 0, which contradicts that there are two distinct additive identities. O The statement u, + 0 + up. which contradicts that 0 is an additive identity. Therefore, the additive identity in a vector space is unique.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove that in a given vector space v, the zero vector is unique. Suppose, by way of contradiction, that there are two distinct additive identities 0 and u. Which of the following statements are then true about the vectors 0 and u,? (Select all that apply.) O The vector 0 + u, does not exist in the vector space V. O The vector 0 + u, is equal to u. O The vector 0 + u, is equal to 0. O The vector 0 + u, is not equal to uo +0 O The vector 0 + u, is not equal to 0. O The vector 0 + u, is not equal to up- Which of the following is a result of the true statements that were chosen and what contradiction then occurs? O The statement u, + 0+ 0, which contradicts that u, is an additive identity. O The statement u, + 0 + 0, which contradicts that u, must have an additive inverse. O The statement u, + 0+ 0 + Ug, which contradicts the commutative property. O The statement u, = 0, which contradicts that there are two distinct additive identities. O The statement u, + 0 + up. which contradicts that 0 is an additive identity. Therefore, the additive identity in a vector space is unique.
Transcribed Image Text:Prove that in a given vector space v, the zero vector is unique. Suppose, by way of contradiction, that there are two distinct additive identities 0 and u. Which of the following statements are then true about the vectors 0 and u,? (Select all that apply.) O The vector 0 + u, does not exist in the vector space V. O The vector 0 + u, is equal to u. O The vector 0 + u, is equal to 0. O The vector 0 + u, is not equal to uo +0 O The vector 0 + u, is not equal to 0. O The vector 0 + u, is not equal to up- Which of the following is a result of the true statements that were chosen and what contradiction then occurs? O The statement u, + 0+ 0, which contradicts that u, is an additive identity. O The statement u, + 0 + 0, which contradicts that u, must have an additive inverse. O The statement u, + 0+ 0 + Ug, which contradicts the commutative property. O The statement u, = 0, which contradicts that there are two distinct additive identities. O The statement u, + 0 + up. which contradicts that 0 is an additive identity. Therefore, the additive identity in a vector space is unique.
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