Linear Algebra

Question

. Indicate whether the statement is true or false. Justify your answer.
(a) If Ax=b has infinitely many solutions, then so does Ax=0

(b) If Ax=b is inconsistent, then Ax=0 has only the trivial solution
(c) The fewest number of hyperplanes in R4 that can intersect in a single point is four
(d) If W is any plane in R3, there is a linear system in three unknowns whose solution set is that plane.
(e) If Ax=b is consistent, then every vector in the solution set is orthogonal to every row vector of A.

Answer 1

(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).