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timkuc
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Why do we need fields (Why do we define fields?)for linear algebra?
timkuc said:Why do we need fields (Why do we define fields?)for linear algebra?
jedishrfu said:and matrix multiplication is anti-commutative.
micromass said:I don't think you meant to say this. See http://en.wikipedia.org/wiki/Anticommutative
That happened to me long ago. Although, for some reason, my friends don't see it that way!jedishrfu said:I guess I'm getting too smart for my own good.
HallsofIvy said:That happened to me long ago. Although, for some reason, my friends don't see it that way!
jedishrfu said:if A and B are two matrices then AB =/= BA unless one is the identity matrix.
micromass said:Sorry to bother you again, but I think this statement is also not accurate. Surely it is possible for two matrices to commute even if both are not the identity?
jedishrfu said:Yes, my mistake its just not commutative ie if A and B are two matrices then AB =/= BA unless one is the identity matrix.
I guess I'm getting too smart for my own good.
In practical applications however, we will always make the field concrete and choose to work in R, C or something else.
Fields are necessary in linear algebra because they provide a structure for performing operations such as addition, subtraction, multiplication, and division on vectors and matrices. Without fields, these operations would not be well-defined and would not follow the properties required for linear algebra.
Fields provide several important properties in linear algebra, including closure, associativity, commutativity, distributivity, and the existence of an identity element and inverse elements for addition and multiplication. These properties allow for the manipulation and solving of equations involving vectors and matrices.
No, not every type of field can be used in linear algebra. The field used must be of characteristic 0, meaning that it does not contain any elements that can be multiplied by 0 to give a non-zero result. Examples of fields that can be used in linear algebra include real numbers, complex numbers, and rational numbers.
Fields are essential for defining vector spaces in linear algebra. A vector space is a collection of objects (vectors) that can be added and multiplied by elements of a field. The field determines the type of numbers that can be used to create the vectors and perform operations on them.
No, fields are not the only mathematical structures used in linear algebra. Other structures such as rings and modules can also be used, but fields are the most commonly used due to their properties that make them suitable for performing operations on vectors and matrices.