Final answer:
To find the basis and dimension of the solution space of the given homogeneous system of linear equations, we can solve the system and determine the variables that make up the solution. In this case, the solution space has dimension 1, and the basis is the single solution (0, 0, 0).
Step-by-step explanation:
To find a basis for the solution space of the given homogeneous system of linear equations, we need to solve the system of equations and determine the variables that make up the solution. The system of equations is:
-x + y + z = 0
3x - y = 0
3x - 5y - 6z = 0
We can start by using the second equation to express y in terms of x: y = 3x. Substituting this into the other equations, we get:
-x + 3x + z = 0
3x - 3x - 6z = 0
Simplifying these equations gives:
2x + z = 0
-6z = 0
From the second equation, we can see that z = 0. Substituting this into the first equation, we get 2x + 0 = 0, which simplifies to x = 0.
Therefore, one solution to the system is x = 0, y = 0, and z = 0.
The solution space of the homogeneous system is the set of all possible solutions. Since we only have one solution, the dimension of the solution space is 1. Therefore, the basis for the solution space is the single solution (0, 0, 0).