# Determine is the set is a real vector space (and say why if it isn't)

• Pi Face
In summary, the set of all nonnegative real numbers is not a vector space because it fails closure under scalar multiplication. For the set of all upper triangular nxn matrices, it is not a vector space because the matrices need to be of the same size for addition to be defined. However, for the set of all upper triangular square matrices, it is a real vector space because the "square" specification means the matrices are all of the same size, allowing for addition and scalar multiplication to preserve the upper triangular property.
Pi Face
-set of all nonnegative real numbers
-set of all upper triangular nxn matrices
-set of all upper triangular square matrices

from the book, the answers are: yes, no(need to be the same size to define addition), and yes

for the set of all nonnegative real numbers, doesn't it fail closure under scalar multiplication? if you multiply any number (except 0) in the set by a negative number, making it a negative number, wouldn't that new number be out of the initial set? thus making it not a vector space?

for upper triangular nxn matrices, why isn't it a vector space? nxn implies they are all the same size right? so when you add two of them together, you get another upper triangular nxn matrix, and multiplying them preserves this as well.

for all upper triagnular square matrices, why is it a real vector space? the set just specifices "square", not the size, so if you have two different sized triaangular matrices then you can't add them, failing closure under addition

anyone have any idea?

I think you are right and the book is wrong, but since it's in a book, I could be missing something

## 1. What is a real vector space?

A real vector space is a mathematical structure that consists of a set of elements (vectors) that can be added together and multiplied by real numbers to produce new vectors. It follows a set of axioms, including closure under addition and scalar multiplication, associativity, and distributivity.

## 2. How do you determine if a set is a real vector space?

To determine if a set is a real vector space, we need to check if it satisfies all the axioms of a vector space. This includes closure under addition and scalar multiplication, existence of an additive identity and inverse, and the distributive and associative properties. If any of these axioms are not satisfied, then the set is not a real vector space.

## 3. What is closure under addition and scalar multiplication?

Closure under addition means that when two vectors are added, the result is also a vector in the same set. Similarly, closure under scalar multiplication means that multiplying a vector by a real number will result in a vector that is also in the same set.

## 4. Can a set be a real vector space if it only satisfies some of the axioms?

No, in order for a set to be a real vector space, it must satisfy all of the axioms. If even one axiom is not satisfied, then the set is not considered a real vector space.

## 5. What are some examples of sets that are not real vector spaces?

Some examples of sets that are not real vector spaces include the set of all polynomials with degree greater than 3, the set of all even integers, and the set of all rational numbers. These sets do not satisfy all the axioms of a real vector space, such as the existence of an additive inverse or closure under scalar multiplication.

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