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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.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.CM: Cumulative Review
Problem 6CM: Let T:R4R2 be the linear transformation defined by T(v)=Av, where A=[10100101]. Find a basis for a...
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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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