Answered: 1) Let A be an m × n matrix. Find all 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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1) Let A be an m × n matrix. Find all b such that the set of solutions to Ax = b is a subspace and prove your claim.

2) Prove or disprove: If T is 1-1, then T^2 is 1-1.

3) Prove or disprove: If T^2 is 1-1, then T is 1-1.

4) Prove or disprove: R^2 with coordinate-wise addition and scaling defined as c(x1, x2) = (x1, 0) is a vector space.

5) Suppose {v1, ..., vn} is a basis for V . Find conditions on c1, ..., cn so that {c1v1, ..., cnvn} is a basis for V and prove your claim.

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$Let w = \{x Ax = b m + n\} claim w ia subspace iff b = 0 pf if b = 0 then w = \{x A x = 0\} Let x, y \in w & α \in If Then Ax = 0, Ay = 0 A (αx + y) = 0 \Rightarrow α x + y \in w$

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