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Determine which (if any) of the following sets U, V,W are subspaces of R". For those that are, justify your answer; for those that are not, show that the set fails to satisfy one of the three defining properties of a subspace by creating a counterexample and justifying that it is indeed a counterexample. 1 U = {A(Bx) |x e R"}, for arbitrary, but fixed matrices A, which is n x m, and B which is m × n. 2 V = {Ax+By | x, y E R"}, for arbitrary, but fixed n x n matrices A and B. 3 W = {x € R" | A²x = x} for A an arbitrary, but fixed n x n matrix.

Determine which (if any) of the following sets U, V,W are subspaces of R". For those that are, justify your answer; for those that are not, show that the set fails to satisfy one of the three defining properties of a subspace by creating a counterexample and justifying that it is indeed a counterexample. 1 U = {A(Bx) |x e R"}, for arbitrary, but fixed matrices A, which is n x m, and B which is m × n. 2 V = {Ax+By | x, y E R"}, for arbitrary, but fixed n x n matrices A and B. 3 W = {x € R" | A²x = x} for A an arbitrary, but fixed n x n matrix.

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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Determine which (if any) of the following sets U, V,W are subspaces of R". For those that are, justify your answer; for those that are not, show that the set fails to satisfy one of the three defining properties of a subspace by creating a counterexample and justifying that it is indeed a counterexample. 1 U = {A(Bx) |x e R"}, for arbitrary, but fixed matrices A, which is n x m, and B which is m × n. 2 V = {Ax+By | x, y E R"}, for arbitrary, but fixed n x n matrices A and B. 3 W = {x € R" | A²x = x} for A an arbitrary, but fixed n x n matrix.
Transcribed Image Text:Determine which (if any) of the following sets U, V,W are subspaces of R". For those that are, justify your answer; for those that are not, show that the set fails to satisfy one of the three defining properties of a subspace by creating a counterexample and justifying that it is indeed a counterexample. 1 U = {A(Bx) |x e R"}, for arbitrary, but fixed matrices A, which is n x m, and B which is m × n. 2 V = {Ax+By | x, y E R"}, for arbitrary, but fixed n x n matrices A and B. 3 W = {x € R" | A²x = x} for A an arbitrary, but fixed n x n matrix.
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