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Problem 10. Show that the mapping A: R² x R² → R defined by ((a, b), (c, d)) → ad – be is bilinear. Show that A(v, w) = -A(w, v) for all v, w in R². Show that the corresponding mapping Cx C- C is given by (z, w) – Im(Zw) and show that this is invariant under rot ation about the origin. That is, if v, w are in R² and v' and w' are their images under rotation by some amount 0, then show that A(v, w) = A(v', w'). Why is %3D this extra-easy with complex numbers?

Problem 10. Show that the mapping A: R² x R² → R defined by ((a, b), (c, d)) → ad – be is bilinear. Show that A(v, w) = -A(w, v) for all v, w in R². Show that the corresponding mapping Cx C- C is given by (z, w) – Im(Zw) and show that this is invariant under rot ation about the origin. That is, if v, w are in R² and v' and w' are their images under rotation by some amount 0, then show that A(v, w) = A(v', w'). Why is %3D this extra-easy with complex numbers?

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Linear Transformations
Section6.2: The Kernewl And Range Of A Linear Transformation
Problem 64E
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Problem 9. Suppose T : R" –→ Rm and S : Rm R* are linear mappings. Show that So T is linear. Problem 10. Show that the mapping A: R? x R² → R defined by ((a, b), (c, d)) – ad – be is bilinear. Show that A(v, w) = –-A(w, v) for all v, w in R?. Show that the corresponding mapping C x C -C is given by (z, w) - Im(Zw) and show that this is invariant under rot ation about the origin. That is, if v, w are in R? and v' and w' are their images under rotation by some amount 0, then show that A(v, w) = A(v', w'). Why is %3D this extra-easy with complex numbers? Problem 11. a) Let u = (1,2, 1) and v = (2, 2,4). Find the line ar combination au + Bv that's as close as possible to w = b) Find the least squares best-fit line for the data set {(1,2), (2, 5), (3, 4), (4, 6)} (plot everything, (3, 2, 1) in R³. of course). Explain your set-up and the connection with the previous part of this problem.
Transcribed Image Text:Problem 9. Suppose T : R" –→ Rm and S : Rm R* are linear mappings. Show that So T is linear. Problem 10. Show that the mapping A: R? x R² → R defined by ((a, b), (c, d)) – ad – be is bilinear. Show that A(v, w) = –-A(w, v) for all v, w in R?. Show that the corresponding mapping C x C -C is given by (z, w) - Im(Zw) and show that this is invariant under rot ation about the origin. That is, if v, w are in R? and v' and w' are their images under rotation by some amount 0, then show that A(v, w) = A(v', w'). Why is %3D this extra-easy with complex numbers? Problem 11. a) Let u = (1,2, 1) and v = (2, 2,4). Find the line ar combination au + Bv that's as close as possible to w = b) Find the least squares best-fit line for the data set {(1,2), (2, 5), (3, 4), (4, 6)} (plot everything, (3, 2, 1) in R³. of course). Explain your set-up and the connection with the previous part of this problem.
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