### 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

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

### Determine Whether the Following Set is a Real Vector Space

Evaluate 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 $ matrices

1. ☐ Closure under Addition
2. ☐ Closure under Scalar Multiplication
3. ☐ Commutativity of Addition
4. ☐ Associativity of Addition
5. ☐ Existence of Zero Vector
6. ☐ Existence of Additive Inverses
7. ☐ Multiplicative Identity
8. ☐ Associativity of Scalar Multiplication
9. ☐ Distributivity over Vector Addition
10. ☐ Distributivity over Scalar Addition
11. ☐ The set is a real vector space.

### Explanation

To 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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