Exploring the Subspace of a Homogeneous System of Linear Equations

[Ryker]

Homework Statement

Suppose you have points of a specific form, say (x, y, 3x + 2y). Show that this set of points is a solution to a homogeneous system of linear equations, hence a subspace.

The Attempt at a Solution

I'm wondering how one is able to go about this. Here's my try, but I'm not sure I'm going about this the right way. I assumed vectors u = (x₁, y₁, 3x₁ + 2y₁) and v = (x₂, y₂, 3x₂ + 2y₂) are two solutions. Next, I said any linear combination of them, say, tu + sv is also a solution to Ax = 0, where x is the solution vector. Then I just expanded the terms and showed that indeed any solution to Ax = 0 as a linear combination of u and v still retains the form (x, y, 3x + 2y).

Is this the proper way of doing this or is there another one? Should I have perhaps instead written down, for example, a₁x + a₂y + a₃(3x + 2y) = 0, expanded it to (a₁ + 3a₃)x + (a₂+2a₃)y = 0, and then resolved this equation to show that the solutions for a's involve parameters?

 

[lanedance]

the points given by (x, y, 3x + 2y) represents a plane through the origin and satisfies 3x + 2y - z = 0 which is a homogenous linear equation

 

[Ryker]

Hmm, yeah, you're right, I didn't think of it that way. But are you sure this is the proper way of formally proving it?

edit: Looking at it again, your solution would then be similar to the second one I proposed, right? I mean, whereas I would get parameters for a's, they could correspond to 3, 2 and -1 in your solution, as well as to multiple other coefficient solutions, such as 6, 4, -2 etc. Could you perhaps elaborate on that a bit?

And could anyone comment on my proposed first solution? Is it wrong, or just another way of going about it?