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Mark each statement True or False. Justify each answer, (if true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case).

Mark each statement True or False. Justify each answer, (if true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case).

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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Mark each statement True or False. Justify each answer, (if true, cite appropriate
facts or theorems. If false, explain why or give a counterexample that shows why
the statement is not true in every case).

f) If an n x n matrix A has n pivot positions, then the reduced echelon form of A is the n x n identity matrix. 9) If A is an m xn matrix, if the equation Ax = b has at least two different solutions, and if the equation Ax = c is consistent, then the equation Ax = c has many solutions. h) If u and v are in Rm, then -u is in Span {u, v}. i) Suppose that v,, v2, and vz are in RS, v2 is not a multiple of v,, and vz is not a linear combination of v, and v2. Then {v1, v2, V3} is linearly independent. j) If A is an m x n matrix with m pivot columns, then the linear transformation x - Ax is a one-to-one mapping.
Transcribed Image Text:f) If an n x n matrix A has n pivot positions, then the reduced echelon form of A is the n x n identity matrix. 9) If A is an m xn matrix, if the equation Ax = b has at least two different solutions, and if the equation Ax = c is consistent, then the equation Ax = c has many solutions. h) If u and v are in Rm, then -u is in Span {u, v}. i) Suppose that v,, v2, and vz are in RS, v2 is not a multiple of v,, and vz is not a linear combination of v, and v2. Then {v1, v2, V3} is linearly independent. j) If A is an m x n matrix with m pivot columns, then the linear transformation x - Ax is a one-to-one mapping.
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