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PROBLEM 2: R¹ Suppose that b is a vector in R2 and 3 is a Let T be a linear transformation R² 1 solution to T(*) = 6. a) Show that the kernel of T is a vector space (i.e. show that it is closed under addition and scalar multiplication). b) Show that if is in ker(7), then s+ is also a solution to T(2) = b. e) Show that if is a solution to T(7)-6, then 5-7 is in ker(7). d) Explain how the picture in problem 1 relates to what you proved in problem 2. EXTRA CREDIT: Do the above problems, but for a linear transformation T: R³-R2 instead. (Note: this means doing some 3D graphing).

PROBLEM 2: R¹ Suppose that b is a vector in R2 and 3 is a Let T be a linear transformation R² 1 solution to T(*) = 6. a) Show that the kernel of T is a vector space (i.e. show that it is closed under addition and scalar multiplication). b) Show that if is in ker(7), then s+ is also a solution to T(2) = b. e) Show that if is a solution to T(7)-6, then 5-7 is in ker(7). d) Explain how the picture in problem 1 relates to what you proved in problem 2. EXTRA CREDIT: Do the above problems, but for a linear transformation T: R³-R2 instead. (Note: this means doing some 3D graphing).

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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PROBLEM 2: Let T be a linear transformation R² R¹. Suppose that b is a vector in R² and 3 is a solution to T() = b. a) Show that the kernel of T is a vector space (i.e. show that it is closed under addition and scalar multiplication). b) Show that if is in ker(7), then s+ is also a solution to T(z) = b. c) Show that if is a solution to T()-b, then 5-7 is in ker(7). d) Explain how the picture in problem 1 relates to what you proved in problem 2. EXTRA CREDIT: Do the above problems, but for a linear transformation T: R³-R2 instead. (Note: this means doing some 3D graphing).
Transcribed Image Text:PROBLEM 2: Let T be a linear transformation R² R¹. Suppose that b is a vector in R² and 3 is a solution to T() = b. a) Show that the kernel of T is a vector space (i.e. show that it is closed under addition and scalar multiplication). b) Show that if is in ker(7), then s+ is also a solution to T(z) = b. c) Show that if is a solution to T()-b, then 5-7 is in ker(7). d) Explain how the picture in problem 1 relates to what you proved in problem 2. EXTRA CREDIT: Do the above problems, but for a linear transformation T: R³-R2 instead. (Note: this means doing some 3D graphing).
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