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Let U, W c V be vector spaces with V = U W and suppose that T E L(V). Define LE L(U,V) by (u) = u, and define T E L(V, U) as follows: If v = u + w with u E U and w e W, then T(v) = u. (a) Explain why 7 is well defined and is in L(V, U). (b) Show that if inTi = Tu, then U is T invariant and rTi = T\1.

Let U, W c V be vector spaces with V = U W and suppose that T E L(V). Define LE L(U,V) by (u) = u, and define T E L(V, U) as follows: If v = u + w with u E U and w e W, then T(v) = u. (a) Explain why 7 is well defined and is in L(V, U). (b) Show that if inTi = Tu, then U is T invariant and rTi = T\1.

Advanced Engineering Mathematics
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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Let U, W c V be vector spaces with V = U W and suppose that T E L(V). Define LE L(U,V) by t(u) = u, and define 7 E L(V, U) as follows: If v = u + w with u E U and w e W, then T(v) = u. (a) Explain why 7 is well defined and is in L(V, U). (b) Show that if inTi = Tu, then U is T invariant and rTi = T\1.
Transcribed Image Text:Let U, W c V be vector spaces with V = U W and suppose that T E L(V). Define LE L(U,V) by t(u) = u, and define 7 E L(V, U) as follows: If v = u + w with u E U and w e W, then T(v) = u. (a) Explain why 7 is well defined and is in L(V, U). (b) Show that if inTi = Tu, then U is T invariant and rTi = T\1.
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