Question

Find a basis for and the dimension of the solution space of the homogeneous system of linear equations -x + y + z = 0, 3x - y = 0, 3x - 5y - 6z = 0. (a) Find a basis for the solution space.

Answer 1

To find a basis for the solution space, we row reduce the augmented matrix of the system to echelon form and identify the pivot columns. The basis for the solution space is the set of vectors corresponding to the free variables. The dimension of the solution space is the number of free variables.

Step-by-step explanation:

To find a basis for the solution space of the given homogeneous system of linear equations, we first write the system in matrix form:

[[-1, 1, 1], [3, -1, 0], [3, -5, -6]][x, y, z] = [0, 0, 0]

Next, we row reduce the augmented matrix to echelon form:

[[-1, 1, 1, 0], [0, 4, 4, 0], [0, 0, -3, 0]]

The pivot columns correspond to variables x and y. The free variable z can be expressed in terms of x and y as z = 0. Therefore, a basis for the solution space is the set {[1, 1, 0], [-1, 0, 1]}.

The dimension of the solution space is 2.