Question: Consider how 'dimension' interacts with 'subset'. Assume U and W are both subspaces of some vector space, and that U W. (a) Prove that

Consider how 'dimension' interacts with 'subset'. Assume U and W are both subspaces of some vector space, and that U ⊂ W.
(a) Prove that dim(U) 6 dim(W).
(b) Prove that equality of dimension holds if and only if U = W.
(c) Show that the prior item does not hold if they are infinite-dimensional.

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