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A vector is defined as I did above. It's a homogeneous tuple, i.e. the Cartesian product of n sets, where all sets are the same. Vector addition plus scalar mul

A vector is defined as I did above. It's a homogeneous tuple, i.e. the Cartesian product of n sets, where all sets are the same. Vector addition plus scalar multiplication are not part of the set. Usually they're part of the definition of a vector space. E.g. (R^2, +, ⋅). But you can construct a vector space out of a set of non-vectors (such as matrices), or out of operations other than vector addition and scalar multiplication, and you can use vectors for purposes other than constructing vector spaces, also without involving either of those operations. For example, to store sequential information.

yakubin 4y ago
Then you're using a definition of vector, not used in any mathematics course. > you can construct a vector space out of a set of non-vectors (such as matrices) A vector is by definition no more and no less than an element of a vector space. Vectors are defined by vector spaces, not the other way around. If you have a vector space whose elements are matrices, then those matrices are vectors. And they will be written in coefficients in a given base as tuples. > out of operations other than vector addition and scalar multiplication You don't "build vectors out of operations like vector addition and scalar multiplication", as in: you don't choose them. You choose the field and dimension, and those operations (vector addition and scalar multiplication) are a consequence. > you can use vectors for purposes other than constructing vector spaces, also without involving either of those operations Again, you don't construct vector spaces out of vectors - there are no vectors without vector spaces. And there are no vector spaces without those operations. But yes, you can use vectors from a given vector space in a greater capacity than just as vectors. An example, which shows the futility of looking at vectors as just tuples: a real number is a vector in the vector space of real numbers over the field of rational numbers.
- fluoridation 4y ago
  >Then you're using a definition of vector, not used in any mathematics course. [...] A vector is by definition no more and no less than an element of a vector space. It's the definition I was given. If you accept that the word "vector" may be given a definition different from the one you give it, you'll need to concede that an std::vector may be a vector. >you don't choose them. You choose the field and dimension, and those operations (vector addition and scalar multiplication) are a consequence. You're using the phrase "vector addition and scalar multiplication" in a different sense than I meant. I was referring to component-wise addition between two vectors and to multiplication between a scalar and the individual components of a vector. You could choose different operations to construct a vector space, as long as they meet the requirements of vector spaces. You used the phrase to refer to the constructing operations of a vector space. So yes, a vector space is indeed constructed out of the operations it is constructed out of. You could have been more charitable in your interpretation of my words, rather than assume I was saying something equivalent to "four-cornered triangle".
bmacho 4y ago
A vector is definitely not defined as you did above. A vector is an element of a vector space, basically anything that you can add together and multiply with an element of a field. It has nothing to do with what you wrote above. [] : https://en.wikipedia.org/wiki/Vector_(mathematics_and_physics) https://en.wikipedia.org/wiki/Vector_(mathematics_and_physic...

yakubin 4y ago

Then you're using a definition of vector, not used in any mathematics course. > you can construct a vector space out of a set of non-vectors (such as matrices) A vector is by definition no more and no less than an element of a vector space. Vectors are defined by vector spaces, not the other way around. If you have a vector space whose elements are matrices, then those matrices are vectors. And they will be written in coefficients in a given base as tuples. > out of operations other than vector addition and scalar multiplication You don't "build vectors out of operations like vector addition and scalar multiplication", as in: you don't choose them. You choose the field and dimension, and those operations (vector addition and scalar multiplication) are a consequence. > you can use vectors for purposes other than constructing vector spaces, also without involving either of those operations Again, you don't construct vector spaces out of vectors - there are no vectors without vector spaces. And there are no vector spaces without those operations. But yes, you can use vectors from a given vector space in a greater capacity than just as vectors. An example, which shows the futility of looking at vectors as just tuples: a real number is a vector in the vector space of real numbers over the field of rational numbers.

fluoridation 4y ago
>Then you're using a definition of vector, not used in any mathematics course. [...] A vector is by definition no more and no less than an element of a vector space. It's the definition I was given. If you accept that the word "vector" may be given a definition different from the one you give it, you'll need to concede that an std::vector may be a vector. >you don't choose them. You choose the field and dimension, and those operations (vector addition and scalar multiplication) are a consequence. You're using the phrase "vector addition and scalar multiplication" in a different sense than I meant. I was referring to component-wise addition between two vectors and to multiplication between a scalar and the individual components of a vector. You could choose different operations to construct a vector space, as long as they meet the requirements of vector spaces. You used the phrase to refer to the constructing operations of a vector space. So yes, a vector space is indeed constructed out of the operations it is constructed out of. You could have been more charitable in your interpretation of my words, rather than assume I was saying something equivalent to "four-cornered triangle".

bmacho 4y ago

A vector is definitely not defined as you did above. A vector is an element of a vector space, basically anything that you can add together and multiply with an element of a field. It has nothing to do with what you wrote above. [] : https://en.wikipedia.org/wiki/Vector_(mathematics_and_physics) https://en.wikipedia.org/wiki/Vector_(mathematics_and_physic...