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I. (10pts) Find all the solutions of the following system of equations (w/o a calculator). T1 + T2 + 3.3 = 0 2.11 + T2 + 4.33 = 1 3.11 + T2 + 5.3 = 2 II. (10pts) T + x2 (a) Show that B is a basis for the space P2 of polynomials of degree less or equal to 2. (b) Write p(x) = 1 + 3.c + 2² as a linear combination of vectors in B. III. (10pts) Let I : R² R² be the identity operator (i.e. I(v) = v for any U € R². Let 1 1 S = { } and T = { -1 D -1 , , }. Find the matrix representation of 1 2 I with respect to (a) S; (b) T; (c) S and T; (d) T and S. IV. (10pts) Let A be the matrix Is A diagonalizable? If yes, find the 0 0 2 0 1 0 0 2 diagonal matrix similar to A. V. (20pts) Let P2 denote the space of polynomials of degree less then or equal to 2. Let T : P2 P2 be defined as T(f) = 2f' + f". (a) Show that T is a linear transformation. (b) Let S = {1,t,t2} Find the matrix representation A of T with respect to the ordered basis S. (c) Find a basis for the kernel of T. and a basis for the image of T. (d) Write the characteristic polynomial of the matrix A from part (b). Find the eigenvalues of A. (e) For each eigenvalue in part (d) find a basis for the corresponding eigenspace. (f) Is A diagonalizable? VI. (16pts) For each statement below answer whether it is TRUE or FALSE. 1. If a vector V is orthogonal to vectors u and W, then it is orthogonal to each vector in the span (u, w). 2. Any set of vectors containing the zero vector is linearly dependent. 3. det = det B. 4. If a linear transformation L : R³ R³ is onto, then it is invertible. 5. If A is a 3 X 5 matrix whose null space has dimension 2, then the equation Ax ==== 0 has infinitely many solutions. 6. Every subspace of Rn has at least one orthonormal basis. 7. If A is a 3 X 3 matrix with eigenvalues / then it is diagonalizable. 8. The determinant of an elementary matrix is always 1.

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