1. Consider the real vector spaces C² and IR2×2, and let f: C² → R²×² defined byf(x, y) = ("'а — da + c _b+d.с — bb+ a),V(x, y) = (a + bi,c + di) E C².(a) Prove that f E L(C², R²×²).(b) Find the matrix M = [f]½ of f with respect to the ordered bases= {(0,1), (0,1 + i), (-i, 1), (1, – 1)} c C² and(c) Determine the Rank and Nullity of the matrix M.(d) Hence, determine whether or not f is invertible.

Answered: Image /qna-images/answer/8d3f0433-f2f7-4f47-be7a-a015e7f3e7e1.jpg

Answered: 1. Consider the real vector spaces C2…

1. Consider the real vector spaces C2 and R2×2, and let f: C2 → R²×² defined by f(x, y) = (4 'а — d a +c b+d. с — b ),v(x, y) = (a + bi, c + di) E C². (a) Prove that f E L(C², R²×²).

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

1. Consider the real vector spaces C² and IR2×2, and let f: C² → R²×² defined by
f(x, y) = ("
'а — d
a + c _b+d.
с — b
b+ a),V(x, y) = (a + bi,c + di) E C².
(a) Prove that f E L(C², R²×²).
(b) Find the matrix M = [f]½ of f with respect to the ordered bases
= {(0,1), (0,1 + i), (-i, 1), (1, – 1)} c C² and
(c) Determine the Rank and Nullity of the matrix M.
(d) Hence, determine whether or not f is invertible.

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