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First off, I would like to do this myself, I'd really like hints on how to proceed so I know where to begin.

  • Let $V_1=\{(a_1, a_2,\ldots , a_n) \mid a_i \in \mathbb C \text{ for } i=1,2,\ldots,n\}$
        Is $V_1$ a vector space over the field of real numbers with the operations of coordinate wise addition and multiplication?
       (Answer: Yes.)
  • Again, Let $V_2=\{(a_1, a_2, \ldots, a_n) \mid a_i \in \mathbb R \text{ for } i=1,2,\ldots,n\}$
       Is $V_2$ a vector space over the field of complex numbers with the operations of coordinate wise addition and multiplication?
       (Answer: No.)

My confusion: We are defining the $n$-tuples on one field and the vector space over another. What is the meaning of that? Can you please differentiate between the two vector spaces in the two problems?

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    $\begingroup$ A vector space is defined on a field $\mathbb{F}$, its subspace could be defined on $S\subset\mathbb{F}$. In this case, if $i\in\mathbb{C}$ and $u=(1,0,0,\cdots)\in V$ (second example) but $iu\not\in V$. $\endgroup$ – user5402 Aug 30 '14 at 20:48
  • $\begingroup$ So for th first example I'll take the scalars from the field $\mathbb R$ and the vectors from the field $\mathbb C$? Did I understand it correctly now? $\endgroup$ – Diya Aug 30 '14 at 20:55
  • $\begingroup$ @Diya It isn't correct to say that you are taking 'vectors from the field $\mathbb C$'. The vectors are merely $n$-tuples formed using the elements of $\mathbb C$. $\endgroup$ – caffeinemachine Aug 30 '14 at 20:59
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There is nothing wrong with defining a vector space over a field $F$ whose elements are $n$-tuples with entries from a field $K$, where $K$ contains $F$.

A simple check will show that all the axioms of vector space are satisfied.

To answer what is the meaning of $\mathbb C^n$ being a vector space over $\mathbb R$, consider the special case of $n=1$. One can visualize the complex numbers as arrows tailed at origin.

For $n>1$ this cannot be visualized and so the question of 'meaning' cannot be answered.

Mathematical objects can be very abstract and 'meaningless'.

As to how to differentiate between the two vector spaces in the two problems: One is a vector space and the other isn't!

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Hint: the scalar multiplication is a function $.:F \times V \rightarrow V$. For every $r \in F$ and every $v \in V$ the product $r.v$ should be a vector in $V$. Now if you take $r=i$ , what happens?

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