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2 1 4. (a) Suppose that a system of equations A = b has a corresponding re- X 3 X 4 duced row echelon form [8 2184] Find the set of solutions to this system of equations. (b) Consider the system of equations x + 2y + 2z =1 2x+ y - 2z = 0 2x - 2y + z =1 (i) Write down the matrix A and vector b such that the above system of equations can be written in matrix form Ax = b. (ii) After reducing the augmented system [A I] to row-echelon form, we obtain 1 2 2 1 0 O 0 1 2 6/9 -3/9 0 0 0 1 2/9 -2/9 1/9 Perform further row operations to find A-1. Please show details of all row operations you use. (iii) Hence, find the solution x to the equation in part (i). (c) Suppose the set of vectors {v1, v2, v3} is linearly dependent. Prove that {v1, v2, v3, v4} is linearly dependent for any vector V4.5. (a) Consider the matrix A 4 CT IA (i) Solve the characteristic equation to show that the eigenvalues of A are * = 7 and 1 = -3. (ii) Hence, find the eigenspaces of A. (b) Suppose that a 2 x 2 matrix has trace 1 and determinant - 4/9. Determine its eigenvalues, showing all working. (c) Suppose that a matrix B satisfies B , = ] , and # - [o']. (i) Give a matrix P which diagonalises B and the corresponding diagonal matrix D. (ii) Use this diagonalisation to prove that B2k = I for any positive integer k. (d) Consider a principal component analysis of some dataset X. The dataset has covariance matrix 3 C = 3 -3 3 3 Show that the vector v = (-1, -1, 1) is a principal component of the dataset X.l 6. (a) Consider the following graph of a probability density function. For m-values greater than zero, the function is dened by f (3:) = 6'23. For :nvalues less than zero, the function is a straight line that meets the maxis at some point a,