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GATE 2026 EE – Question 49

Engineering Mathematics · Linear Algebra: Matrix Algebra · 2 marks · Multiple choice

Which one of the following statements is ALWAYS correct about a collection of $p$ column vectors, each having $n$ real-valued entries?

  1. If $p>n$, then the column vectors must be linearly dependent
  2. If $p>n$, then the column vectors must be linearly independent
  3. If $p=n$, then the column vectors must be orthogonal
  4. If $p<n$, then the column vectors must be linearly independent

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Correct answer: (A) If $p>n$, then the column vectors must be linearly dependent

Explanation

The space $\mathbb R^n$ has dimension $n$, so no more than $n$ vectors can be linearly independent: with $p>n$ vectors they must be dependent. For $p=n$ or $p<n$ the vectors could be dependent or non-orthogonal (e.g. two equal vectors), so C and D are not always true.