Thread: Solutions of Linear Systems II

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.

[3] Consider the two systems represented by the augmented matrices.
[ 1 1 -5 3; 1 0 -2 1; 2 -1 -1 0] [ 1 1 -5 -9; 1 0 -2 -3; 2 -1 -1 0]
If the first system is known to be consistent, explain why the second system is also consistent. Do not perform any row operations.

[4] The vectors x1 and x2 are solutions of the linear system Ax=b. Is the vector 2x1-3x2 also a solution? Why or why not?

In [3], the first three columns of the two matrices are equal, right? What about the last column of the two matrices, are they somehow related?

As for [4], recall that the matrix A works as a _linear_ transformation. You want to check whether or not x=2x1-3x2 solves Ax=b, when you know that Ax1=b and Ax2=b.