Understanding "Linearity" in Hawking & Ellis' Subspace of T_p

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I'm reading Hawking and Ellis, and on p.16 they say that

"The subspace of T_p defined by \langle \omega,x \rangle=(constant) for a given one-form \omega, is linear."

But in what sense is this true? For if the constant is non-zero, the 0 of T_p is not in the subspace, nor does it satisfy the usual linearity condition.

By analogy with euclidean space, the points r \cdot a = d form a plane, but not technically a subspace of the original space (since it is "shifted"). Am I missing something?
 
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No, I don't think that you are missing anything, especially considering the sentence that follows the sentence which you quoted.
 
Thanks!

Turns out that's the least of my difficulties in this book, but it's quite interesting.
 
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