So I'm reading these notes about differential geometry as it relates to general relativity. It defines a tensor as being, among other things, a linear scalar function, and soon after it gives the following equation as an example of this property of linearity:(adsbygoogle = window.adsbygoogle || []).push({});

T(aP + bQ, cR + dS) = acT(P, R) + adT(P, S) + bcT(Q, R)+bdT(Q, S)

whereTis the tensor function, P, Q, R, and S are vectors, and a, b, c, and d are scalar coefficients.

Now I can follow the above leap from left hand side to right hand side as far as:

T(aP + bQ, cR + dS) =T(aP, cR + dS) +T(bQ, cR + dS) =T(aP, cR) +T(aP, dS) +T(bQ, cR) +T(bQ, dS)

but I don't quite understand the reasoning behind how the coefficients get outside of the function brackets. Somehow I managed to get a bachelors in physics without ever taking a single linear algebra course, so I'm a little bit stumped.

Can anyone here give me a hand with this? Any help would be greatly appreciated.

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# GR Math: how does tensor linearity work?

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