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(4.1) Let V, V, and Vs be vector spaces over a field F. Let T: VV and SV₂V₁ be linear transformations. (a) Show that ST: VV is a linear transformation. (b) Show that ker TC ker (ST) and im (ST) C im S. (c) Deduce that ST can only be bijective if T is injective and S is surjective. (3) (4) (3)

(4.1) Let V, V, and Vs be vector spaces over a field F. Let T: VV and SV₂V₁ be linear transformations. (a) Show that ST: VV is a linear transformation. (b) Show that ker TC ker (ST) and im (ST) C im S. (c) Deduce that ST can only be bijective if T is injective and S is surjective. (3) (4) (3)

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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(4.1) Let V, V, and Vs be vector spaces over a field F. Let T: VV and SV₂V₁ be linear transformations. (a) Show that ST: VV is a linear transformation. (b) Show that ker TC ker (ST) and im (ST) C im S. (c) Deduce that ST can only be bijective if T is injective and S is surjective. (3) (4) (3)
Transcribed Image Text:(4.1) Let V, V, and Vs be vector spaces over a field F. Let T: VV and SV₂V₁ be linear transformations. (a) Show that ST: VV is a linear transformation. (b) Show that ker TC ker (ST) and im (ST) C im S. (c) Deduce that ST can only be bijective if T is injective and S is surjective. (3) (4) (3)
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