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 it, thank you~.

(1) To give an example, Ax=b is the only solution.

(2) 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, if m = n, then det(A) ≠ 0 and A will be invertible. Then x = A^-1 b is a unique solution. For example, let A =

1	1
2	3

and b = (3,5)^T. Apparently, the columns of A are linearly independent so that A is invertible. Also, A^-1 =

3	-1
-2	1

The unique solution is x = A^-1 b =(4,-1)^T.

(2). If equation Ax = b has only one solution, then we must have m = n. If m ≠n, this equation will have infinite solutions.