Let c_1, c_2, ..., c_n be scalars, not all zero, such that sum_{i=1}^{n} c_i * a_i = 0 where a_i are column vectors in R^n. Consider the set of linear equations Ax = b where A = [a_1, a_2, ..., a_n] and b = sum_{i=1}^{n} a_i. The set of equations has (J_n denotes a n-dimensional vector of all 1s): A. a unique solution at x = J_n where J_n = [1,1,...,1]^T B. no solution C. infinitely many solutions D. finitely many solutions

GATE 2017 · Engineering Mathematics · System of Equations · medium

Answer: The system has infinitely many solutions.

1. Establish that A is singular: Since c_1,...,c_n are not all zero and sum c_i*a_i = 0, the columns of A are linearly dependent. Therefore A is singular, meaning null(A) has dimension >= 1, i.e., there exists a nonzero vector v such that Av = 0.
2. Identify a particular solution: b = a_1 + a_2 + ... + a_n = A * [1,1,...,1]^T = A * J_n. So x* = J_n is a particular solution of Ax = b.
3. Conclude infinitely many solutions: Since Av = 0 for any null-space vector v, and x* = J_n is a particular solution, every x = J_n + alpha*v (for alpha in R) is also a solution. The null space has dimension >= 1, giving infinitely many solutions.