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17. Let V be a vector space over F and let W be a subspace of V. Let T be a linear operator on V and suppose that W is invariant under T. (a) Show that if u+W=v+W then T(u)+W=T(v)+W; in particular the function T: V/W → V/W given by the formula T(u + W) = T(u) + W is well defined. (b) Show that the function T from part (a) is in fact a linear operator on the quotient vector space V/W over F.

17. Let V be a vector space over F and let W be a subspace of V. Let T be a linear operator on V and suppose that W is invariant under T. (a) Show that if u+W=v+W then T(u)+W=T(v)+W; in particular the function T: V/W → V/W given by the formula T(u + W) = T(u) + W is well defined. (b) Show that the function T from part (a) is in fact a linear operator on the quotient vector space V/W over F.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CR: Review Exercises
Problem 47CR: Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
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17. Let V be a vector space over F and let W be a subspace of V. Let T be a linear operator on V and suppose that W is invariant under T. (a) Show that if u+W=v+W then T(u)+W=T(v)+W; in particular the function T: V/W → V/W given by the formula T(u + W) = T(u) + W is well defined. (b) Show that the function T from part (a) is in fact a linear operator on the quotient vector space V/W over F.
Transcribed Image Text:17. Let V be a vector space over F and let W be a subspace of V. Let T be a linear operator on V and suppose that W is invariant under T. (a) Show that if u+W=v+W then T(u)+W=T(v)+W; in particular the function T: V/W → V/W given by the formula T(u + W) = T(u) + W is well defined. (b) Show that the function T from part (a) is in fact a linear operator on the quotient vector space V/W over F.
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