The GATE Grind

GATE 2017 CS – Question 13

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Let $c_1, \ldots, c_n$ be scalars, not all zero, such that $\sum_{i=1}^{n} c_i a_i = 0$ where $a_i$ are column vectors in $\mathbb{R}^n$.

Consider the set of linear equations

$$Ax = b$$

where $A = [a_1, \ldots, a_n]$ and $b = \sum_{i=1}^{n} a_i$. The set of equations has

  1. a unique solution at $x = J_n$ where $J_n$ denotes a $n$-dimensional vector of all 1
  2. no solution
  3. infinitely many solutions
  4. finitely many solutions

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Show answer and explanation

Correct answer: (C) infinitely many solutions

Explanation

Since $b = A \cdot J_n$, the vector of all ones is a solution, so the system is consistent. The scalars $c_i$, not all zero, give a nonzero vector $c$ with $Ac = 0$, so the columns are dependent. Then $x = J_n + tc$ is a solution for every $t$, so there are infinitely many solutions.