Suppose T: P3→R³ is a linear transformation whose action is defined by 2a+2b-4c+2d T(ax3 +bx2+cx+d) = a+b-2c+d -3a-2b+c+2d Determine whether T is one-to-one and/or onto. If it is not one-to-one, show this by pro under T. If I is not onto, show this by providing a vector in R³ that is not in the image and x as the variable for polynomials, e.g. 5x^2-2x+1 T is not one-to-one: 0 0 T(0)= 0 and T(0) = 0 0 0 T is not onto: 0 0 is not the image of any polynomial under T. 0
Suppose T: P3→R³ is a linear transformation whose action is defined by 2a+2b-4c+2d T(ax3 +bx2+cx+d) = a+b-2c+d -3a-2b+c+2d Determine whether T is one-to-one and/or onto. If it is not one-to-one, show this by pro under T. If I is not onto, show this by providing a vector in R³ that is not in the image and x as the variable for polynomials, e.g. 5x^2-2x+1 T is not one-to-one: 0 0 T(0)= 0 and T(0) = 0 0 0 T is not onto: 0 0 is not the image of any polynomial under T. 0
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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Show two examples for the polynomials
Transcribed Image Text:Suppose T: P3 R³ is a linear transformation whose action is defined by
2a+2b-4c+2d
T(ax3 +bx2+cx+d) = a+b-2c+d
-3a-2b+c+2d
Determine whether T is one-to-one and/or onto. If it is not one-to-one, show this by pro
under T. If I is not onto, show this by providing a vector in R³ that is not in the image
and x as the variable for polynomials, e.g. 5x^2-2x+1
T is not one-to-one:
0
0
T(0)= 0 and T(0) = 0
0
0
T is not onto:
0
0 is not the image of any polynomial under T.
0
Transcribed Image Text:Suppose T: P3R³ is a linear transformation whose action is defined by
2a+2b-4c+2d
a+b-2c+d
T(ax3 +bx²+cx+d) =
-3a-2b+c+2d
Determine whether T is one-to-one and/or onto. If it is not one-to-one, show this by providing two polynomials that have the same image
under T. If T is not onto, show this by providing a vector in R³ that is not in the image of T. Use the '^' character to indicate an exponent
and x as the variable for polynomials, e.g. 5x^2-2x+1
T is one-to-one
T is onto
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