(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)
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 24EQ
Question
![(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)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc4e4a331-de9f-4f44-bf90-60b228d3a50a%2Fe07513bb-1125-4453-a3d2-346fde6cde49%2Frttv24r_processed.jpeg&w=3840&q=75)
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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