4. a. Let V be a finite-dimensional vector space and let T : V → V be linear. Show that if T is one-to-one, then T is onto. b. Give an example of a function f: RR which is one-to-one but not onto. Why does this not contradict the Rank-Nullity Theorem?

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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4.
a. Let V be a finite-dimensional vector space and let T : V → V be linear. Show that if
T is one-to-one, then T is onto.
b. Give an example of a function f: RR which is one-to-one but not onto. Why does
this not contradict the Rank-Nullity Theorem?
Transcribed Image Text:4. a. Let V be a finite-dimensional vector space and let T : V → V be linear. Show that if T is one-to-one, then T is onto. b. Give an example of a function f: RR which is one-to-one but not onto. Why does this not contradict the Rank-Nullity Theorem?
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