What is the Precise Definition of a “Complex Vector Space”?
2019年7月13日 · Specifically, Axler (and Wolfram, and others) define a “complex vector space” as a vector space in which the field is the complex numbers. According to this definition, the complex numbers over the real numbers are not a complex vector space, but the complex numbers over the complex numbers are a complex vector space.
Complex vector spaces - Mathematics Stack Exchange
If that gives you the entire vector space, you're done: that one vector form a basis. If the multiples of that vector don't give you the entire vector space, pick some vector you don't get as a multiple of the first one, and look at all the linear combinations of the two vectors you have selected - the "span" of the two vectors.
What is the "standard basis" for fields of complex numbers?
$\begingroup$ What is not clear (to me, at least) from your question is that you consider $\Bbb C^2$ as a real or complex vector space. As a complex vector space it has dimension $2$, as a real vector space it has dimension $4$. $\endgroup$ –
Linear Algebra: Complex Vector Space is also a Real Vector Space
2020年1月23日 · Extending a real vector space into a complex vector space using a linear map 2 A function in a Real Vector Space V (where V is the set of all complex-valued functions f on the real line ), which is NOT real-valued.
linear algebra - Orthogonal vectors in complex vector space ...
2021年10月29日 · When you specify a vector space, you always specify two things: a set and a field. When we take our set to be the complex numbers $ \mathbf{C} $, two possible choices for our field are $ \mathbf{R} $ and $ \mathbf{C} $. So we may think of $ \mathbf{C} $ as a vector space with either real scalars or complex scalars. The choice is important when ...
$\\mathbb{C}$ is a one-dimensional complex vector space. What …
The complex numbers as a vector space over the field of real numbers is of dimension $2$.
Norm of Complex Vector - Mathematics Stack Exchange
2016年2月24日 · The dot product for complex vectors is given by ... The norm of a complex vector is given by $$ \|(w,z ...
Can the field of complex numbers be regarded as a vector space?
2017年7月7日 · "The field C of complex numbers may be regarded as a vector space over the field ℝ of real numbers. More generally, let F be the field of real numbers and let V be the set of n-tuples $\ α=(x_1,..,x_n)$ where $\ x_1,..,x_n$ are complex numbers.
What is the dot product of complex vectors?
2017年10月6日 · which would work on both real and complex cases. form: maintained obviously. This complex "dot product" is sometimes called a Hermitian form. This specific separate term serves as a way to make it clear that it might not comply with the usual definition of a dot product, if you don't generalize that definition as shown above.
Complex vector space, but with a real inner product (Hilbert sub …
2021年6月7日 · Say I have a complex-valued vector space $\mathbb{C}^n$ with the following inner product: $$ \langle u,v\rangle=u^Tv $$ If one picks only vectors that have real entries, and only allow linear transformations from $\mathbb{R}^n\to \mathbb{R}^n$. Then, can we say that this complex vector space embeds a real Hilbert sub-space, in some sense?