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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

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 3EQ
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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: 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: 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
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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