Linear algebra: Finding a linear system with a subspace as solution set

[sphlanx]

Homework Statement

We are given a subspace of R^3 that is produced by the elements: (2,6,2) abd (6,2,2). We are asked to find (if any) a homogeneous linear system that has this subspace as solution set.

Homework Equations

The Attempt at a Solution

1)The subspace is 2 dimensional so the solution set must have 2 parameters. Also, given the elements that produce the subspace, i guess we want a system with 3 variables and 2 equations.

No clue after that :S

 

[HallsofIvy]

Homework Statement

We are given a subspace of R^3 that is produced by the elements: (2,6,2) abd (6,2,2). We are asked to find (if any) a homogeneous linear system that has this subspace as solution set.

Homework Equations

The Attempt at a Solution

1)The subspace is 2 dimensional so the solution set must have 2 parameters. Also, given the elements that produce the subspace, i guess we want a system with 3 variables and 2 equations.

No clue after that :S

Yes, that's right. You want to equations, say ax+ by+ cz= P and dx+ ey+ fz= Q that are both satisfied by (2,6,2) and (6,2,2). That is, you must have the four equations 2a+ 6b+ 2c= P, 2d+ 6e+ 2f= Q, 6a+ 2b+ 2c= P, and 6d+ 2e+ 2f= Q. That gives you four equations to solve for 8 numbers, but, of course there are many sets of equations that will satisfy this problem. Solve for four of the variables in terms of the other four, then choose whatever numbers you please for those four.