# Linear Algebra: Understanding max(x,y) in x \oplus y

• harvellt
In summary, the conversation is about a new operation \oplus defined as the maximum of two integers. The question is whether this operation satisfies the properties of a vector space, such as associativity and commutativity. The conversation also discusses the definition of a vector space and the properties that must be satisfied.
harvellt
Not sure if this is the best place for this, its not an entire problem I am having trouble with but a small part of one.
I am working on linear algebra and I can't find a good explanation for
$$x \oplus y= max(x,y)$$

What or of max is it? Additive, multiplicative?

Thank you so much been trying to do my own research but can't seem to find it.

It looks like the author is defining a new operation $\oplus$ that is not either addition or multiplication. The result of applying $\oplus$ to two integers is, by definition, the greater of the two integers. Presumably, you are then asked to prove whether or not this operation satisfies certain properties (commutivity, associativity, etc.).

Does that help?

Petek

yep exactly that is very helpful! So you think that $$max(1,2)=2$$? for example?
The full problem defines scalar multiplication normally and then gives that operation over Reals and asks if it is a vector space.

harvellt said:
yep exactly that is very helpful! So you think that $$max(1,2)=2$$? for example?
The full problem defines scalar multiplication normally and then gives that operation over Reals and asks if it is a vector space.

Correct, so $1 \oplus 2 = 2$.

Petek

"Addtion" in a vector space has to satisfy:
1) Associative. Is $a\oplus(b\oplus c)= (a\oplus b)\oplus c$ for all numbers, a, b, and c?
2) Commutative. Is $a\oplus b= b\oplus c$?
3) Distributive. Is $a(b\oplus c)= ab\oplus ac$?
4) Additive identity. Is there some "e" such that $$\displaystyle a\oplus e= a$$ for all a?
5) Additive inverses. For every number a, is there a number b such that a\oplus b= e, where e is as in (4)?

For example, $a\oplus(b\oplus c)= a\oplus max(b,c)$= max(a, max(b,c))= max(a,b,c).

## 1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their representations in vector spaces. It is used to solve problems related to systems of linear equations, transformations, and geometric concepts.

## 2. What is x ⊕ y in linear algebra?

x ⊕ y, also known as the direct sum of x and y, is a fundamental concept in linear algebra that represents the combination of two vectors or matrices. It is used to define operations such as addition and multiplication in vector spaces.

## 3. How is max(x,y) defined in x ⊕ y?

In linear algebra, max(x,y) is defined as the maximum value between x and y. In the context of x ⊕ y, it represents the maximum element in the direct sum of x and y. This can be calculated by comparing the elements in each vector and choosing the larger value for each corresponding position.

## 4. What is the significance of understanding max(x,y) in x ⊕ y?

Understanding max(x,y) in x ⊕ y is important because it allows us to determine the maximum value in a combined vector or matrix. This can be useful in a variety of applications, such as optimization problems and data analysis.

## 5. How can I improve my understanding of max(x,y) in x ⊕ y?

The best way to improve your understanding of max(x,y) in x ⊕ y is to practice solving problems involving direct sums and maximum values. You can also refer to textbooks and online resources for further explanations and examples. Collaborating with peers and seeking guidance from a teacher or tutor can also be helpful.

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