### Determine Whether the Following Set is a Real Vector SpaceEvaluate whether the given set is a real vector space, and select the properties that do not hold if it is not a real vector space.#### The set of all upper triangular \( n \times n \) matrices1. ☐ Closure under Addition2. ☐ Closure under Scalar Multiplication3. ☐ Commutativity of Addition4. ☐ Associativity of Addition5. ☐ Existence of Zero Vector6. ☐ Existence of Additive Inverses7. ☐ Multiplicative Identity8. ☐ Associativity of Scalar Multiplication9. ☐ Distributivity over Vector Addition10. ☐ Distributivity over Scalar Addition11. ☐ The set is a real vector space.### ExplanationTo check if the set of all upper triangular \( n \times n \) matrices forms a real vector space, the following properties must hold:1. **Closure under Addition:** The sum of any two upper triangular matrices must also be an upper triangular matrix.2. **Closure under Scalar Multiplication:** Any upper triangular matrix multiplied by a scalar should remain an upper triangular matrix.3. **Commutativity of Addition:** For any two upper triangular matrices \(A\) and \(B\), \(A + B = B + A\).4. **Associativity of Addition:** For any three upper triangular matrices \(A\), \(B\), and \(C\), \((A + B) + C = A + (B + C)\).5. **Existence of Zero Vector:** The set must include the zero matrix, which is an upper triangular matrix.6. **Existence of Additive Inverses:** For every upper triangular matrix \(A\), there must exist an upper triangular matrix \(-A\) such that \(A + (-A) = 0\).7. **Multiplicative Identity:** This does not apply directly because we are dealing with vector spaces, not necessarily fields or rings.8. **Associativity of Scalar Multiplication:** For any scalar \(c\) and any upper triangular matrices \(A\) and \(B\), \(c(A + B) = cA + cB\).9. **Distributivity over Vector Addition:** For any scalars \(c\) and \(d\) and any upper triangular matrix \(A\

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Determine whether the following set is a real vector space (and select what fails if it is not). The set of all upper triangular n xn matrices Closure under Addition Closure under Scalar Multiplication O Commutativity of Addition Associativity of Addition Existence of Zero Vector Existence of Additive Inverses O Multiplicative Identity Associativity of Scalar Multiplication Distributivity over Vector Addition O Distributivity over Scalar Addition O The set is a real vector space.

Determine whether the following set is a real vector space (and select what fails if it is not). The set of all upper triangular n xn matrices Closure under Addition Closure under Scalar Multiplication O Commutativity of Addition Associativity of Addition Existence of Zero Vector Existence of Additive Inverses O Multiplicative Identity Associativity of Scalar Multiplication Distributivity over Vector Addition O Distributivity over Scalar Addition O The set is a real vector space.

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 un. Which of the following statements are then true about the vectors 0 and u.? (Select all that apply ) O The vector 0 + u, is not equal to u, +0 O The vector 0 + u, does not exist in the vector space V. O The vector 0 + u, is not equal to 0. O The vector 0 + u, is equal to un O The vector 0 + u, is equal.to 0 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, which contradicts that u, must have an additive inverse. O The statement u, + 0 + 0 + u, which contradicts the commutative property. O The statement u, = 0, which contradicts that there are two distinct additive identities. O The statement u, + 0 + 0, which contradicts that u, is an additive identity. The statement u, + 0 + u. which contradicts that 0…