a) Let F be the field of complex numbers and let T be the function from F3 into F3 defined by T(x1, X2, X3) = (x, – x2 + 2x3,2x1 + x2, - X1 - 2 x2 + 2x3 ) i. Verify that T is a linear transformation. ii. If (a, b, c) is a vector in F3, what are the conditions on a, b and c that the vector be in the range of T? What is the rank of T? iii. What are the conditions on a, b and c that the vector (a, b,c) be in the null space of T? What is the nullity of T?
a) Let F be the field of complex numbers and let T be the function from F3 into F3 defined by T(x1, X2, X3) = (x, – x2 + 2x3,2x1 + x2, - X1 - 2 x2 + 2x3 ) i. Verify that T is a linear transformation. ii. If (a, b, c) is a vector in F3, what are the conditions on a, b and c that the vector be in the range of T? What is the rank of T? iii. What are the conditions on a, b and c that the vector (a, b,c) be in the null space of T? What is the nullity of T?
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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Transcribed Image Text:Q-2 (a) Let F be the field of complex numbers and let T be the function from F3 into F3 defined by
T(x1, x2, X3) = (X1 -x2 +2x3, 2x1 + X2,-x1 - 2 x2 + 2x3)
i. Verify that T is a linear transformation.
If (a, b, c) is a vector in F3, what are the conditions on a, b and c that the vector be
in the range of T? What is the rank of T?
What are the conditions on a, b and c that the vector (a, b, c) be in the null space of
T? What is the nullity of T?
ii.
iii.
(b) Let T be the linear operator on R3 defined by
T(x1, x2, X3) = (3x1, X1 - X2, 2x1 + x2 +x3)
Is T invertible? If so, find a rule for T-1 like the one which defines T.
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