State whet her each statement is true or false and provide a short justification for yourclaim (a short proof if you think the statement is true or a counter example if you thinkit is false).(a)Let Apxn, n > 1, be a matrix such that RREF(A) has only one pivotcolumn. Let i be a nonzero vector in Col(A). Then is an eigenvector for A.(b) Let T :V → W be a linear transformation, and suppose that w e V is alinear combination of the vectors õ1, ..., Un in V. Then T(w) is a linear combinationof the vectors T(71),...,T(ūn).

Solution: (a) The given statement is TRUE. Proof: For matrix An×n, let the one pivot column is ith…

State whether each statement is true or false and provide a short justification for your claim (a short proof if you think the statement is true or a counter example if you think it is false). (a) Let Apxn, n > 1, be a matrix such that RREF(A) has only one pivot column. Let be a nonzero vector in Col(A). Then is an eigenvector for A. Let T : V → W be a linear transformation, and suppose that w e V is a linear combination of the vectors ī1,..., ữn in V. Then T(w) is a linear combination of the vectors T(u1),...,T(ūn). (b)

State whether each statement is true or false and provide a short justification for your claim (a short proof if you think the statement is true or a counter example if you think it is false). (a) Let Apxn, n > 1, be a matrix such that RREF(A) has only one pivot column. Let be a nonzero vector in Col(A). Then is an eigenvector for A. Let T : V → W be a linear transformation, and suppose that w e V is a linear combination of the vectors ī1,..., ữn in V. Then T(w) is a linear combination of the vectors T(u1),...,T(ūn). (b)

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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Question

State whet her each statement is true or false and provide a short justification for your
claim (a short proof if you think the statement is true or a counter example if you think
it is false).
(a)
Let Apxn, n > 1, be a matrix such that RREF(A) has only one pivot
column. Let i be a nonzero vector in Col(A). Then is an eigenvector for A.
(b) Let T :V → W be a linear transformation, and suppose that w e V is a
linear combination of the vectors õ1, ..., Un in V. Then T(w) is a linear combination
of the vectors T(71),...,T(ūn).

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