Linear Algebra - Associative property

In summary, the problem is to prove the associative property for the operation x*y=(x+y)-⌊x+y⌋ with elements x and y in the set G={x∈R|0≤x<1}. By manipulating the nested floor functions, it can be shown that the statement holds true. The condition where x+y+z<x+y is not applicable in this task.
  • #1
Karamata
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Homework Statement


[tex]G=\{x\in R|0\leq x<1\}[/tex] and for some [tex]x,y\in G[/tex] define [tex]x*y=\{x+y\}=x+y-\lfloor x+y \rfloor[/tex]

Homework Equations


The Attempt at a Solution


I want to proof Associative property:
[tex]x*(y*z)=(x*y)*z \Leftrightarrow x*(y+z-\lfloor y+z \rfloor)=(x+y-\lfloor x+y \rfloor)*z [/tex]
[tex]\Leftrightarrow x+y+z-\lfloor y+z \rfloor-\lfloor {x+y+z-\lfloor y+z \rfloor}\rfloor=x+y+z-\lfloor x+y \rfloor-\lfloor x+y+z-\lfloor x+y \rfloor\rfloor[/tex]
[tex] \Leftrightarrow\lfloor y+z \rfloor+\lfloor {x+y+z-\lfloor y+z \rfloor}\rfloor=\lfloor x+y \rfloor+\lfloor x+y+z-\lfloor x+y \rfloor\rfloor[/tex]
What now?
 
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  • #2
The nested floor functions can be taken outside of their encapsulating floor functions since they are integers
 
  • #3
[tex]\lfloor x+y+z-\lfloor y+z \rfloor \rfloor=\lfloor x+y+z\rfloor-\lfloor y+z \rfloor[/tex]
Right? That seems well.

Can we do that if [tex]x+y+z<x+y[/tex] (ok, that isn't true in this task), and if [tex]x+y+z\geq 0[/tex] and [tex]y+z\geq 0[/tex]


Sorry for bad English.
 

1. What is the associative property in linear algebra?

The associative property in linear algebra states that the order in which matrix multiplication is performed does not affect the final result. In other words, when multiplying three or more matrices, the grouping of the multiplication does not matter as long as the same matrices are involved.

2. How is the associative property used in linear algebra?

The associative property is used to simplify and manipulate complex matrix expressions. By rearranging the order of operations, it can make calculations more efficient and easier to solve.

3. What is an example of the associative property in action?

For example, when multiplying three matrices A, B, and C, the associative property states that (AB)C = A(BC) = ABC. This means that the order in which the multiplications are performed does not change the final result.

4. Does the associative property hold true for all matrix operations?

Yes, the associative property holds true for all matrix operations, including addition, subtraction, and multiplication. This is because the properties of matrix operations are based on the fundamental properties of numbers, which also include the associative property.

5. Are there any other properties in linear algebra that are related to the associative property?

Yes, the associative property is closely related to the commutative and distributive properties. The commutative property states that the order of operands does not affect the result, while the distributive property is used to expand and simplify expressions involving matrix multiplication and addition.

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