Jacobians in 7 DoF serial manipulators

Anyone here particularly familiar with Jacobians and their use?

I am working on a 7 degree of freedom system. I believe/recall from classes that Jacobians are limited to containing 6 degrees of freedom, that with any [6 x n] system, n<7 so that the Jacobian can never exceed [6 x 6] and still be useful. I recall the reasoning being issues dimension mismatching with any matrix mathematics involving matrices of [6 x n], where n>6. I also recall that something like a [6 x7] matrix would have issues with its rank, and so would run into singularities more frequently – but I am far less sure of this point.

Instead, I recall that when you have a 7-DoF system, one must use some kind of kinematics decoupling and arrange their joints in such a way that the decoupling is possible.

But someone I know, who is quite knowledgeable in robotics, is questioning whether this is true or not. They are not sure one way or another, but they do question whether a standard Jacobian has an upper limit on containing 6 individual joints. Anyone know the answer to this debate, or where I might be able to learn more?

I’ve been re-reading Spong 2006, but have never been a fan on its chapter about Jacobians (does a good job covering how to calculate a Jacobian, less-so when it comes to their exact usage, limitations, and general context of their usage)

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