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# Linear Independence on a Set of Complex Vectors

## Published by Jas, the Physicist

My current interest is flows: flowing on a beat, flowing along a manifold, flows on the electromagnetic spectrum, flow curves, flows of a vector field, the flow of time, time as currency, the flow of electrons: currency, like electric currents, electric daisies, current interest rates, current events, swimming against the current, fluid flow, flux and divergence... you know... flowers. View more posts

Linear independence in the complex plane doesn’t look the same as the real number line, for obvious reasons. Lines aren’t circles, right? That was basically what all of covering spaces attempted to prove when they created the local homeomorphism of R onto S1. Lines aren’t not circles. Lines locally approximate circles.

So now, when you think of linear dependence of two vectors in the complex plane, draw me a picture of what you see.

Circles of different periods, or periodically intersecting.

I know this isn’t a picture. I’ve now moved up to S^n and can’t vizualize as well.

I have a picture for s^3 but how can I see s^3 inside of s^3?

Edit. I meant R3.