Find a basis for the solution space of the following homogeneous system of linear equations. 3x1 + 9x2 + 30x3 + 21x4 = 0 5x4 = 0 16x4 = 0 X1 - 3x2 - 8x3 + 2x1 + 12x2 + 38x3 + O {(1, 0, 1, 6), (0, 3, 9, 0), (1, 0, 0, 1)} O {(0, 0, 1, 6), (1, 3, 9, 1)} {(1, 0, 0, 0), (0, 3, 0, 0), (0, 0, 10, 7)} O {(-1, -3, 1, 0), (-18, -1, 0, 3)} O {(1, 0, 0, 0), (0, 3, 0, 0), (0, 0, 10, 0), (0, 0, 0, 7)}

Solve the following homogeneous system of linear equations: x1+x2+x3-2x4 = 0 -3x1-3x2+2x3-9x4 = 0 -3x1-3x2-2x3+6x4 = 0 -x1-x2-3x3+7x4 = 0 -2x1-2x2-x3-2x4 = 0 If the system has no solution, demonstrate this by giving a row-echelon form of the augmented matrix for the system. You can resize a matrix (when appropriate) by clicking and dragging the bottom-right corner of the matrix. The system has infinitely many solutions The system has no solution The system has a unique solution The system has infinitely many solutions x2 = 0 + s0 X3

The nonlinear system has six real solutions. 4x0 x₁ + x₂ = X0X3 xo - 2x₁ + 3x₂ = x₂x3 -xo + 3x₁ - 2x₂ = X₁ X3 x² + x² + x² = 1 (a) Show that if (x0, X₁, X2, X3)T is a solution, then (-X0, −X₁, −X₂, X3)¹ is also a solution.

Find A, x, and b such that Ax = b corresponds to the given linear system. (Do not perform any row operations.) -10 2x₁ + 9x₂ - 4x3 = -X₁ - 3x₂ + 8x3 = 4 A = X = b = ↓ ↑ ↓ ↑ →