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3. Suppose T: M22 → P3 is a linear transformation whose action on a basis for M22 is as follows: 3. T T [1] =−x³ + x²-3x+2 = x T T -3x²-x-6 == = -4x³8x² - 10x - 16 Determine a matrix, Ar that takes the matrix entry vectors to the coefficient vectors of the polynomials. Then determine a basis for the kernel and image of T. Is T one-to-one and/or onto? If it is not one-to-one, show this by providing two matrices that have the same image under T. If T is not onto, show this by providing a polynomial in P3 that is not in the image of T and show that it is not in the image of T.

3. Suppose T: M22 → P3 is a linear transformation whose action on a basis for M22 is as follows: 3. T T [1] =−x³ + x²-3x+2 = x T T -3x²-x-6 == = -4x³8x² - 10x - 16 Determine a matrix, Ar that takes the matrix entry vectors to the coefficient vectors of the polynomials. Then determine a basis for the kernel and image of T. Is T one-to-one and/or onto? If it is not one-to-one, show this by providing two matrices that have the same image under T. If T is not onto, show this by providing a polynomial in P3 that is not in the image of T and show that it is not in the image of T.

Advanced Engineering Mathematics
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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3. Suppose T: M22 P3 is a linear transformation whose action on a basis for M22 is as follows: 3- = T T = -x³ + x²-3x+2 = X T T :-3x² - x-6 = -4x³8x² 10x - 16 Determine a matrix, Ar that takes the matrix entry vectors to the coefficient vectors of the polynomials. Then determine a basis for the kernel and image of T. Is T one-to-one and/or onto? If it is not one-to-one, show this by providing two matrices that have the same image under T. If T is not onto, show this by providing a polynomial in P3 that is not in the image of T and show that it is not in the image of T.
Transcribed Image Text:3. Suppose T: M22 P3 is a linear transformation whose action on a basis for M22 is as follows: 3- = T T = -x³ + x²-3x+2 = X T T :-3x² - x-6 = -4x³8x² 10x - 16 Determine a matrix, Ar that takes the matrix entry vectors to the coefficient vectors of the polynomials. Then determine a basis for the kernel and image of T. Is T one-to-one and/or onto? If it is not one-to-one, show this by providing two matrices that have the same image under T. If T is not onto, show this by providing a polynomial in P3 that is not in the image of T and show that it is not in the image of T.
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