Linear Algebra - Determining a Solution to AX = B

[KingKai]

Homework Statement

Assume that

$$A \left( \begin{array}{c} 1 \\ -1 \\ 2 \end{array} \right) = 0 = A \left( \begin{array}{c} 2 \\ 0 \\ 3\end{array} \right)$$

and that AX = B has a solution $$Xo= \left( \begin{array}{c} 2 \\ -1 \\ 3 \end{array} \right)$$

Find a two parameter family of solutions to AX = B

Homework Equations

AX = B

$$A \left( \begin{array}{c} 1 \\ -1 \\ 2 \end{array} \right) = 0 = A \left( \begin{array}{c} 2 \\ 0 \\ 3\end{array} \right)$$

The Attempt at a Solution

Representing coefficient columns of matrix A as a (A₁ , A₂ , A₃) for corresponding n-vectors of x,

A₁ - A₂ + 2A₃ = 2A₁ + 3A₃

A₂ = -A₁ - A₃

not sure where to go from here or really what i am doing.

 

[vela]

This is more of a conceptual problem. Think about things like the homogeneous solution, particular solution, null space, etc.