Name: CSU ID: Homework 7 October 23, 2015 1. Find the standard matrix for the stated composition in <3 (a) The rotation of 30◦ about the x-axis, followed by a rotation of 30◦ about the z-axis, followed by a contraction with factor k = 1/4. (b) A reflection about the xy-plane followed by a reflection about the xz-plane, followed by an orthogonal projection onto the yz-plane. 2. Let B = {M1 , M2 , M3 , M4 } be the ordered basis given below. Find the coorinates of the vector M = I2×2 relative to the ordered basis B. " M1 = 1 0 2 0 # " , M2 = −1 5 0 2 # " , M3 = 4 6 8 3 # " , M4 = 3 −4 6 3 # 3. Let B = 1 + 2x2 , −1 + 5x + 2x3 , 4 + 6x + 8x2 + 3x3 , 3 − 4x + 6x2 + 3x3 . Show that B is a basis for P 3 , polynomials of degree less than or equal to 3. For the polynomial p(x) = 1 + x + x2 + x3 , determine [p]B . 6. Find the eigenvalues and eigenvectors of A where 1 3 0 0 A= 4 1 0 0 −2 7. Find the eigenvalues and eigenvectors of A 0 −3 5 4 −10 A = −4 0 0 4 8. The matrix A given below has eigenvalues λ = 2, −5, 3. (a) For each eigenvalue, find the eigenvector by bringing λI − A to RREF. The eigenvector should be defined with integers. (b) Define a matrices S and D such that S −1 AS = D is a diagonal matrix with the ordered eigenvalues given in (a). Do Not Solve for S −1 5 −4 −2 27 14 A = −2 5 −50 −26 9. Show that for A in the previous problem that the characteristic polynomial of A, pA (x), satisfies pA (A) = 03×3

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# Name: Homework 7 CSU ID: October 23, 2015