Linear Combinations, Homogeneous Systems
3.57. Write v as a linear combination of u1, u2, u3, where
(a) v = (4, -9, 2), u1 = (1, 2, -1), u2 = (1, 4, 2), u3 = (1, -3, 2);
(b) v = (1, 3, 2), u1 = (1, 2, 1), u2 = (2, 6, 5), u3 = (1, 7, 8);
(c) v = (1, 4, 6), u1 = (1, 1, 2), u2 = (2, 3, 5), u3 = (3, 5, 8).
3.58. Let u1 = (1, 1, 2), u2 = (1, 3, -2), u3 = (4, -2, -1) in R^3. Show that u1, u2, u3 are orthogonal, and write v as a linear combination of u1, u2, u3, where (a) v = (5, -5, 9), (b) v = (1, -3, 3), (c) v = (1, 1, 1). (Hint: Use Fourier coefficients.)
3.59. Find the dimension and a basis of the general solution W of each of the following homogeneous systems:
(a) x - y + 2z = 0, 2x + y + z = 0, 5x + y + 4z = 0
(b) x + 2y - 3z = 0, 2x + 5y + 2z = 0, 3x - y - 4z = 0
(c) x + 2y + 3z + t = 0, 2x + 4y + 7z + 4t = 0, 3x + 6y + 10z + 5t = 0
3.60. Find the dimension and a basis of the general solution W of each of the following systems:
(a) x1 + 3x2 + 2x3 - x4 - x5 = 0, 2x1 + 6x2 + 5x3 + x4 - x5 = 0, 5x1 + 15x2 + 12x3 + x4 - 3x5 = 0
(b) 2x1 - 4x2 + 3x3 - x4 + 2x5 = 0, 3x1 - 6x2 + 5x3 - 2x4 + 4x5 = 0, 5x1 - 10x2 + 7x3 - 3x4 + 18x5 = 0
Echelon Matrices, Row Canonical Form
3.61. Reduce each of the following matrices to echelon form and then to row canonical form:
(a) [[1, 1, 2], [2, 4, 9], [1, 5, 12]], (b) [[1, 2, -1, 2, 1], [2, 4, 1, -2, 5], [3, 6, 3, -7, 7]], (c) [[2, 4, 2, -2, 5, 1], [3, 6, 2, 2, 0, 4], [4, 8, 2, 6, -5, 7]]
3.62. Reduce each of the following matrices to echelon form and then to row canonical form:
(a) [[1, 2, 1, 2, 1, 2], [2, 4, 3, 5, 5, 7], [3, 6, 4, 9, 10, 11], [1, 2, 4, 3, 6, 9]], (b) [[0, 1, 2, 3], [0, 3, 8, 12], [0, 0, 4, 6], [0, 2, 7, 10]], (c) [[1, 3, 1, 3], [2, 8, 5, 10], [1, 7, 7, 11], [3, 11, 7, 15]]
3.63. Using only 0's and 1's, list all possible 2 x 2 matrices in row canonical form.
3.64. Using only 0's and 1's, find the number n of possible 3 x 3 matrices in row canonical form.