Please, help me with a step by step and detailed solutions Let f: V W be any linear map of vector spaces over a field K (a) Suppose f is surjective. Describe all possible injective linear maps s: W -> V so that fos Iw (the identity map on W, that is, f(s(u)) Hint: Fix a basis for W and note that, for two such maps s1 and s2, f(s1 (u))u f(82(u)) iff s1(u) - 82(u) E Ker(f).) (b) Suppose f is injective. Describe all possible surjective linear maps T: W V so that T o f = Iv u for all u E W).
Please, help me with a step by step and detailed solutions Let f: V W be any linear map of vector spaces over a field K (a) Suppose f is surjective. Describe all possible injective linear maps s: W -> V so that fos Iw (the identity map on W, that is, f(s(u)) Hint: Fix a basis for W and note that, for two such maps s1 and s2, f(s1 (u))u f(82(u)) iff s1(u) - 82(u) E Ker(f).) (b) Suppose f is injective. Describe all possible surjective linear maps T: W V so that T o f = Iv u for all u E W).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Please, help me with a step by step and detailed solutions
Let f: V W be any linear map of vector spaces over a field K
(a) Suppose f is surjective.
Describe all possible injective linear maps s: W -> V so that
fos Iw (the identity map on W, that is, f(s(u))
Hint: Fix a basis for W and note that, for two such maps s1 and s2,
f(s1 (u))u f(82(u)) iff s1(u) - 82(u) E Ker(f).)
(b) Suppose f is injective. Describe all possible surjective linear maps
T: W V so that T o f = Iv
u for all u E W).
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