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zcd

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_{1},a

_{2},...):a

_{i}∈C} , could the vector space be over the field R so that I only take scalars from the reals?

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- Thread starter zcd
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zcd

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mathwonk

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well,yes, but what is really your question?

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zcd

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However, every (finite dimensional space) is isomorphic to F

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Fredrik

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It's easy to show that this vector space is 3-dimensional, and that the set of Pauli spin matrices is a basis. With an appropriate choice of inner product, this basis is orthonormal.

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mathwonk

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Are you asking whether, say the complex numbers can be considered as a vector space over the real numbers? Then the answer is yes, but in writing them down as "tuples" one would then use real numbers to represent them, not complex numbers. So the phrase "entries inside the vectors" is the part that is too imprecise for me to understand perfectly.

I.e. when you say this, I implicitly assume you mean entries in a basis representation, which do come from the base field, but that is apparently not what you meant.

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Fredrik

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When he says "vector" he means "tuple", not "member of a vector space". So he's asking if ℂSo the phrase "entries inside the vectors" is the part that is too imprecise for me to understand perfectly.

I'm pretty sure he means entries in theI.e. when you say this, I implicitly assume you mean entries in a basis representation,

Not necessarily. I don't see why anyone would want to do this, but you can give ℂThen the answer is yes, but in writing them down as "tuples" one would then use real numbers to represent them, not complex numbers.

The example I used in my previous post is much more interesting. It's a 3-dimensional vector space over ℝ, and the basis I mentioned looks like this:

[tex]\sigma_1=\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix},\quad \sigma_2=\begin{pmatrix}0 & -i\\ i & 0\end{pmatrix},\quad \sigma_3=\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}[/tex]

So an arbitrary member can be written as

[tex]x=\sum_{k=1}^3 x_k\sigma_k=\begin{pmatrix}x_3 & x_1-ix_2\\ x_1+ix_2 & -x_3\end{pmatrix}[/tex]

Now, the tuple that you and I would associate with this vector and this basis is [itex](x_1,x_2,x_3)[/itex], but the OP's phrase "inside the vector" refers to what's inside the matrix on the right-hand side.

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zcd

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Sorry for not being clear with my question. I'm asking whether the field a particular vector space is over has any bearing on what's inside a vector from the vector space, or are those two separate matters?I'm sorry, this question is so imprecise as not to make sense to me, but the others seem to know what you mean.

So what you're doing is using a vector space of 2x2 matrices with complex entries, but since the scalars used for scalar multiplication are all reals the vector space is over the field of reals?The example I used in my previous post is much more interesting. It's a 3-dimensional vector space over ℝ, and the basis I mentioned looks like this:

[tex]\sigma_1=\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix},\quad \sigma_2=\begin{pmatrix}0 & -i\\ i & 0\end{pmatrix},\quad \sigma_3=\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}[/tex]

So an arbitrary member can be written as

[tex]x=\sum_{k=1}^3 x_k\sigma_k=\begin{pmatrix}x_3 & x_1-ix_2\\ x_1+ix_2 & -x_3\end{pmatrix}[/tex]

Now, the tuple that you and I would associate with this vector and this basis is [itex](x_1,x_2,x_3)[/itex], but the OP's phrase "inside the vector" refers to what's inside the matrix on the right-hand side.

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Fredrik

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Exactly. One of the things that makes this example interesting is that we don't have the option to choose the field of scalars to be ℂ, because if A is a member of the set, iA isn't. (iA)*=-iA*=-iA≠iA.So what you're doing is using a vector space of 2x2 matrices with complex entries, but since the scalars used for scalar multiplication are all reals the vector space is over the field of reals?

It's the "inside a vector" part of your statement that's unclear. "Vector" means "member of a vector space", not "member of FSorry for not being clear with my question. I'm asking whether the field a particular vector space is over has any bearing on what's inside a vector from the vector space, or are those two separate matters?

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Landau

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zcd

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Thanks for the clarification!

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