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).
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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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c741b4f-da7f-48ba-b84e-2097952246cb%2F8d3f0433-f2f7-4f47-be7a-a015e7f3e7e1%2Fznev2rd_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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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