4 ms·

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)

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".

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".