Solutions of Linear Systems I

Write a short paragraph to answer each of the following questions about solutions of systems of linear equations. You should not perform any calculations, but instead base your explanations on the appropriate properties.

[1] One solution of the homogeneous linear system

x + 2y + z + 3w = 0

x - y + + w = 0

y - z + 2w = 0

is x=-2, y=-1, z=1, ans w=1. Explain why x=4, y=2, z=-2, and w=-2 must also be a solution. Do not perform any row operations.

[2] The vectors x1 and x2 are solutions of the homogeneous linear system Ax=0. Explain why the vector 2x1-3x2 must also be a solution.