How to find basis and parameters ?

Question

The subspace U={(x,y,z)∈R|x−ay=b,x−az=b} is given and I am having a trouble with finding its basis, dimension and the parameters a and b.

Answer 1

The problem provides insufficient information for finding the parameters a, b: this problem is asking you to determine a basis for U as a function of the parameters as well as determining the dimension of U as a function of the parameters.

We can decompose the solution as the sum of a particular solution to the general system and the space of solutions to the homogeneous system.

Pick some value for x (say, x = 0) and solve for y, z as functions of a,b.

(x = 0) -> (y = -b/a, z = -b/a)

This gives you a particular solution as (-b/a)*(0, 1, 1).

Then solve the homogeneous system:

Ax = 0, where A is determined from the equations describing the subspace U, and x is solved for algebraically.

I don't know how to typeset it for this forum, so I will give you the solution explicitly. Thus the general solution is:

x = (-b/a)*(0,1,1) + c1*(a, 1, 1)

Hence U is one-dimensional and the basis vector is (a, 1, 1).

*Note: this was an AFFINE Algebra problem, so you need to consider translations together with linear transformations.