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.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section: Chapter Questions
Problem 1RQ
Related questions
Question
please answer all
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 4 images
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning