Let L: U V be a linear function, and let W U be a subspace
Question:
(a) Prove that Y = {L[w] | w ∈ W} ⊂ rng L ⊂ V is a subspace of the range.
(b) Prove that dim Y ≤ dim W. Conclude that a linear transformation can never increase the dimension of a subspace.
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a First Y rng L since every y Y can be written as y Lw for some w W U and so y rng L If y 1 Lw 1 and ...View the full answer
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