# How to visualise complex vector spaces of dimension 2 and above

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• pellis
In summary: This is a very abstract way of thinking about complex numbers, and it may not be helpful if you want to actually do algebra and analysis with C2. However, it may be helpful if you want to understand how to think about complex numbers and how they relate to geometry.
pellis
TL;DR Summary
Although the algebra of complex vector spaces (Dimension >= 2) makes sense to me, I’m not sure about how to imagine the geometry.
According to e.g. Keith Conrad (https://kconrad.math.uconn.edu/blurbs/ choose Complexification) If W is a vector in the vector space R2, then the complexification of R2, labelled R2(c), is a vector space W⊕W, elements of which are pairs (W,W) that satisfy the multiplication rule for complex vectors. Therefore R2(c) is isomorphic to the 2D complex space C2.

The elements of R2(c), i.e. of C2, can be thought of, by what appears to be a slight abuse of notation, as (W+iW).

And the elementary algebra of complex vector spaces treats elements as linear combinations of basis vectors having complex coefficients (i.e. the basis vectors need not themselves be complex).

Does this mean that:

- Every vector in C2 has its own complex plane?

OR

- Only the basis vectors in C2 are complex planes?

OR

- Something different from either of the above.

It all depends on what you want to do, i.e. in which branch of mathematics you are.

A finite-dimensional vector space over any field is always the same. You have basis vectors, then its multiples of any scalar, real, finite, or complex, and all you can reach from there with vector additions.

Only the scalar multiples might be a bit unusual in the finite or complex case. It all starts with what you want to do. A vector space for me is ##\mathbb{F}v_1+\ldots+\mathbb{F}v_n## regardless of what ##\mathbb{F}## is.

The complexification suggests that you want to consider real and imaginary parts separately. Why?

If ##W## is a real vector space, then ##W\otimes_\mathbb{R}\mathbb{C}## is its complexification.

Last edited:
Klystron

I'm a chemist trying to make more sense of spinors and especially C2 where 2-component spinors such as Pauli and Weyl spinors "live". I've long been aware of this, but not how to think of C2 geometrically.

You wrote "A vector space for me is Fv1+…+Fvn regardless of what F is." And I'm fine with that, but what does it look like for n = 2 and F = C?

Trivially, one visualises C1 as the standard Argand diagram. But if you extend C1 to C2, can the result be thought of as two real axes and two imaginary axes? Or two intersecting complex planes? Or two real axes sharing one imaginary axis, or a space in which every vector has "its own" complex plane as the (W+iW) in my question above, seems to suggest?

As a chemist I'm used to building a visual sense of something, even for abstractions. But I'm also well aware of the quote about the difficulty of understanding spinors, from the recently deceased Sir Michael Atiyah.

Hopefully, this makes my query clearer.

If we draw a line, then it is a one-dimensional, real vector space.

But ##\mathbb{C}^2## has a four-dimensional realization, something we cannot imagine. At least most of us.

So all we can do is visualize ##\mathbb{C}^2## as a plane, with the strange property that ##\vec{v}## and ##i\cdot \vec{v}## point in the same direction. However, we cannot draw ##i\cdot\vec{v}.## As soon as we try we have real numbers.

The imagination of the complex numbers as a real plane is a crutch. One that fails as soon as we make analysis with it. That is why I like the abstract algebraic visualization: All ##\mathbb{C}\cdot \vec{v}## point into the same direction. We can do algebra and analysis with it, but we have to make painful compromises if we try to draw something.

And as a chemist, you will know that you cannot draw an electron. A ball is a crutch, too. Nevertheless, we can work well with electrons. Even if visualizations are always a compromise.

Thank you for that.

I had wondered whether one could consider slices or projections of C2 to gain more geometric insight.

But for now I'll try to think about "C2 as a plane, with the strange property that v→ and i⋅v→ point in the same direction."

## 1. How do I represent a vector in a 2-dimensional space?

In a 2-dimensional space, a vector can be represented by two coordinates, typically denoted as (x,y). These coordinates indicate the magnitude and direction of the vector in relation to the x and y axes.

## 2. How can I visualize a vector space of dimension 2 and above?

One way to visualize a vector space of dimension 2 and above is by using a graph or coordinate system. Each vector can be represented as a point on the graph, with its coordinates indicating its direction and magnitude. This allows for a visual representation of the relationships and transformations within the vector space.

## 3. What are some techniques for understanding complex vector spaces?

One technique for understanding complex vector spaces is by breaking them down into smaller, more manageable parts. This can involve looking at individual vectors and their relationships, as well as examining the overall structure of the space. Another technique is to use visual aids, such as graphs or diagrams, to help conceptualize the space.

## 4. How do I determine the dimension of a vector space?

The dimension of a vector space is determined by the number of linearly independent vectors it contains. This can be found by using the rank-nullity theorem, which states that the dimension is equal to the rank (number of linearly independent vectors) plus the nullity (number of vectors that can be written as a linear combination of other vectors).

## 5. What is the difference between a basis and a spanning set in a vector space?

A basis is a set of linearly independent vectors that can be used to represent any vector in a vector space. A spanning set, on the other hand, is a set of vectors that can be used to create any vector in the space through linear combinations. While a basis is unique and minimal, a spanning set may contain redundant or unnecessary vectors.

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