22 ms·

So... are you going to answer them?

dominotw 13y ago
Gilbert Strang explains clearly why we need change of basis with an example of image compression here, http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-31-change-of-basis-image-compression/ http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-...
- infogulch 13y ago
  How about "Is Euclidean space the only vector space out there?" And "Do vectors have to be lists of numbers?" Perhaps with some examples?
  - mej10 13y ago
    http://en.wikipedia.org/wiki/Examples_of_vector_spaces http://en.wikipedia.org/wiki/Examples_of_vector_spaces
  - defen 13y ago
    > "Do vectors have to be lists of numbers?" No. They just have to satisfy the vectors axioms (greatly simplifying, you can add vectors together and multiply them by scalars , and they behave as you would expect - associative, commutative, distributive, etc as appropriate.) So, in this regard, functions can be vectors, if you define addition to be pointwise addition. For example, consider the set of continuous functions on the unit interval [0, 1]. Add two together and you get another continuous function on the unit interval, multiply by a scalar and you get another continuous function on the unit interval, etc.
  - j2kun 13y ago
    It turns out that, if you restrict to spaces of finite dimension, then Euclidean space is the only vector space out there, up to your choice of labelling the basis elements and up to a certain linear transformation. In this sense, vectors can always be written as lists of numbers, but no, they don't have to be this in general. For example, you could have a vector space of smooth functions. It's infinite dimensional, and its elements (its vectors) are functions, not tuples of numbers.
    - hacman 13y ago
      You must also add the condition that your scalars are real numbers. There are many finite dimensional vector spaces that are nothing like Euclidean space: any finite extension of a finite field, for example. This comes from Galois theory, but is not just abstract nonsense. One application to computing is if your scalars are only 0 or 1 and all operations are done mod 2, then you have a framework for doing error correcting codes, among other things. We call this set of scalars either the finite field of size 2 or the Galois field of size 2, aka GF(2).
      - j2kun 13y ago
        Yes, what I should have said is: if k is an algebraically closed field then n-dimensional vector spaces over k are isomorphic to k^n. Unless I've had too much to drink this evening.

Gilbert Strang explains clearly why we need change of basis with an example of image compression here, http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-31-change-of-basis-image-compression/ http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-...

infogulch 13y ago
How about "Is Euclidean space the only vector space out there?" And "Do vectors have to be lists of numbers?" Perhaps with some examples?
- mej10 13y ago
  http://en.wikipedia.org/wiki/Examples_of_vector_spaces http://en.wikipedia.org/wiki/Examples_of_vector_spaces
- defen 13y ago
  > "Do vectors have to be lists of numbers?" No. They just have to satisfy the vectors axioms (greatly simplifying, you can add vectors together and multiply them by scalars , and they behave as you would expect - associative, commutative, distributive, etc as appropriate.) So, in this regard, functions can be vectors, if you define addition to be pointwise addition. For example, consider the set of continuous functions on the unit interval [0, 1]. Add two together and you get another continuous function on the unit interval, multiply by a scalar and you get another continuous function on the unit interval, etc.
- j2kun 13y ago
  It turns out that, if you restrict to spaces of finite dimension, then Euclidean space is the only vector space out there, up to your choice of labelling the basis elements and up to a certain linear transformation. In this sense, vectors can always be written as lists of numbers, but no, they don't have to be this in general. For example, you could have a vector space of smooth functions. It's infinite dimensional, and its elements (its vectors) are functions, not tuples of numbers.
  - hacman 13y ago
    You must also add the condition that your scalars are real numbers. There are many finite dimensional vector spaces that are nothing like Euclidean space: any finite extension of a finite field, for example. This comes from Galois theory, but is not just abstract nonsense. One application to computing is if your scalars are only 0 or 1 and all operations are done mod 2, then you have a framework for doing error correcting codes, among other things. We call this set of scalars either the finite field of size 2 or the Galois field of size 2, aka GF(2).
    - j2kun 13y ago
      Yes, what I should have said is: if k is an algebraically closed field then n-dimensional vector spaces over k are isomorphic to k^n. Unless I've had too much to drink this evening.