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The Lie Algebra is equipped with a bracket notation, and this bracket produces skew symmetric matrices.

I know that there exists Lie Groups, one of which is SO(3).

And I know that by exponentiating a skew symmetric matrix, I obtain a rotation matrix.

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First, can someone edit the following and say it correctly and with precision (I am not certain I am saying it correctly and the phrasing matters to me)

"If one exponentiatiates a MEMBER of the Lie ALgebra, one obtains a member of the Lie Group SO(3)."

Could you fix that sentence for me?

Second...

I get the idea the angular velocity matrices are skew symmetric. I get the idea that rotations are orthogonal. But does this act of exponentiating a skew symmetric matrix imply a one to one correspondence between a specific angular velocity and a rotation? What does that imply?

WHAT IS IT about the philosophy (is that a good word? theory? )of exponentiating a matrix that enables this connection between Lie Groups and Lie Algebras? What is it about the fact that we are using a function whose derivative is itself, that causes rotations and angular velocities to be connected (is connected a good word?)?

I know that there exists Lie Groups, one of which is SO(3).

And I know that by exponentiating a skew symmetric matrix, I obtain a rotation matrix.

-----------------

First, can someone edit the following and say it correctly and with precision (I am not certain I am saying it correctly and the phrasing matters to me)

"If one exponentiatiates a MEMBER of the Lie ALgebra, one obtains a member of the Lie Group SO(3)."

Could you fix that sentence for me?

Second...

I get the idea the angular velocity matrices are skew symmetric. I get the idea that rotations are orthogonal. But does this act of exponentiating a skew symmetric matrix imply a one to one correspondence between a specific angular velocity and a rotation? What does that imply?

WHAT IS IT about the philosophy (is that a good word? theory? )of exponentiating a matrix that enables this connection between Lie Groups and Lie Algebras? What is it about the fact that we are using a function whose derivative is itself, that causes rotations and angular velocities to be connected (is connected a good word?)?

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