Section 2.3: 12(c) only! 12. Let V, W, and Z be vector spaces, and let T: VW and U: W → Zbe linear.(a) Prove that if UT is one-to-one, then T is one-to-one. Must U alsobe one-to-one?(b) Prove that if UT is onto, then U is onto. Must T also be onto?(c) Prove that if U and T are one-to-one and onto, then UT is also.

Note: We will be solving part (c) only as asked.Let and are vector spaces.Let and be linear.We…

Answered: Let V, W, and Z be vector spaces, and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Let V, W, and Z be vector spaces, and let T: V → W and U: W → Z be linear. (a) Prove that if UT is one-to-one, then T is one-to-one. Must U also be one-to-one? (b) Prove that if UT is onto, then U is onto. Must T also be onto? (c) Prove that if U and T are one-to-one and onto, then UT is also.