Question

A linear system of equations Ax=b is known, where A is a matrix of m by n size, and the column vectors of A are linearly independent of each other. Please answer the following questions based on this assumption, please explain all questions, thank you~.

(1) Please explain why this system has at most one solution.

(2) To give an example, Ax=b is no solution.

(3) According to the previous question, what kind of inference can be made to the size of A at this time? (What is the size of m and n,please explain also it thanks.)

Answer 1

We presume that A has real entries. If not, we have to replace Rⁿ by Cⁿ.

(1). Since A is a mxn matrix with linearly independent columns, hence the columns of A form a basis for Rⁿ ( as dim(Rⁿ) = n) . Therefore, regardless of the choice of b, it is a linear combination of the columns of A. Hence the equation Ax = b will be consistent. Further, since A is a mxn matrix, it is not invertible so that the equation Ax = b will have infinite solutions.

(2). There is no question of the equation Ax = b not having any solution as b belongs to the column space of A.

(3). If m = n, then A is a square matrix with n linearly independent columns so that A is invertible. Then x = A^-1 b is a unique solution. However, if m ≠ n, then A is not invertible. Further, since regardless of the choice of b, it is a linear combination of the columns of A, hence the equation Ax = b will be consistent and will have infinite solutions.