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Prove that in a given vector space V, the zero vector is unique.

Prove that in a given vector space V, the zero vector is unique.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 34EQ
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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 statement are then true about the vectors 0 and u,? (Select all that apply.) The vector 0 + u, is not equal to 0. The vector 0 + u, is not equal to u, + 0. The vector 0 + u, is equal to 0. The vector 0 + does not exist in the vector space V. O The vector 0 + u, is equal to u̟. O The vector 0 + u, is not equal to u,. 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 + u,, which contradicts the commutative property. O The statement u, + 0 + un, which contradicts that 0 is an additive identity. The statement + 0 + 0, which contradicts that u, must have an additive inverse. The statement u, = 0, which contradicts that there are two distinct additive identities. The statement u, + 0 + 0, which contradicts that u, 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 statement are then true about the vectors 0 and u,? (Select all that apply.) The vector 0 + u, is not equal to 0. The vector 0 + u, is not equal to u, + 0. The vector 0 + u, is equal to 0. The vector 0 + does not exist in the vector space V. O The vector 0 + u, is equal to u̟. O The vector 0 + u, is not equal to u,. 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 + u,, which contradicts the commutative property. O The statement u, + 0 + un, which contradicts that 0 is an additive identity. The statement + 0 + 0, which contradicts that u, must have an additive inverse. The statement u, = 0, which contradicts that there are two distinct additive identities. The statement u, + 0 + 0, which contradicts that u, is an additive identity. Therefore, the additive identity in a vector space is unique.
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