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Mathematics
Linear and Abstract Algebra
How to visualise complex vector spaces of dimension 2 and above
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[QUOTE="fresh_42, post: 6579895, member: 572553"] 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. [/QUOTE]
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Forums
Mathematics
Linear and Abstract Algebra
How to visualise complex vector spaces of dimension 2 and above
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