Question

Consider a system of n linear equations with m variables, which has a coefficient matrix A of size nxm (a) Filling the blank: total number of variables = number of leading variables + (b) Using (a) to prove that if the system has exactly one solution, then rank (A) = m (c) Using (a) to prove that if the system has infinitely many solutions, then rank (A) < m

Answer 1

(a) The total number of variables in the system is equal to the number of leading variables plus the number of free variables.

(b) If the system has exactly one solution, then it means that all variables can be uniquely determined from the system of equations. This implies that every column of A must have a pivot position, which means that the rank of A is equal to m.

(c) If the system has infinitely many solutions, then it means that there are at least one or more free variables in the system. This implies that there cannot be a pivot position in every column of A, which means that the rank of A must be less than m.

Explanation: