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The outer diameter of a glass jar and the inner diameter of its iron lid are both 725 mm at room temperature (20.0°C). What will be the size of the mismatch between the lid and the jar if the lid is briefly held under hot water until its temperature rises to 50.0°C, without changing the temperature of the glass?

We can ask about the algebra of the ‘+’ operation.

(a) Is it commutative, is W1 +W2 = W2 +W1?

(b) Is it associative, is (W1 +W2) +W3 = W1 + (W2 +W3)?

(c) Let W be a subspace of some vector space. Show that W +W = W.

(d) Must there be an identity element, a subspace I such that I+W = W+I = W for all subspaces W?

(e) Does left-cancellation hold: if W1 +W2 = W1 +W3 then W2 = W3? Right cancellation?

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