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6. DETAILS LARLINALG8 6.1.014.EP. Consider the following function. T: R2 → R3, T(x, y) = (2x2, 3xy, 2y2) Find the following images for vectors u = (44, 42) and v = (V , V2) in R2 and the scalar c. (Give all answers in terms of Uq, U2, V1, V2, and c.) T(u) = T(v) = T(u) + T(v) = T(u + v) = CT(u) = T(cu) = Determine whether the function is a linear transformation. O linear transformation O not a linear transformation
8. DETAILS LARLINALG8 7.2.007.EP. Find the characteristic equation and eigenvalues of the matrix and a basis for each of the corresponding eigenspaces. 8 A A-:-) - (-:-?] the characteristic equation 2² – 92=0 the eigenvalues (Enter your answers from smallest to largest.) (02, ) = ( 0,9 = a basis for each of the corresponding eigenspaces B1 = 1 B, = { -2.1 For the matrix A, use the basis of the eigenspaces to find a nonsingular matrix P such that p-1AP is diagonal. P = 11 Verify that P-1AP is a diagonal matrix with the eigenvalues on the main diagonal. p-AP = 11
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