(a) If Ax=b has infinitely many solutions, then for any two solutions x_1 and x_2 we have A(x_1-x_2) = 0 and thereby Ax=0 has infinitely many solutions.

(b) That is not true, an easy example would be A=(0) and b arbitrary, but not zero.

(c) Assume, we have three or less hyperplanes intersecting in just one point. Every hyperplane is given by the solutions of a linear equation. Hence, we have a consistent (as they are intersecting in at least this point) system of at most three linear equations. What do we know about such a system?

(d) Yes, we can definitely construct this. We find the set of vectors orthogonal to W. This leads to a system of equations defining the plane which is "parallel" to W and includes the origin. Now we just have to shift W.

(e) That's false. Example: A=(1 & 1//1&1), b = (1,1). Then (1,0) is a solution of the system, yet not orthogonal to (1,1).

Olga J.

(c) a linear system is consistent if and only if b is in the column space of A ? I am not sure:(

12/29/12