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GATE 2017 EC – Question 13

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Consider the following statements about the linear dependence of the real valued functions $y_1 = 1$, $y_2 = x$ and $y_3 = x^2$, over the field of real numbers.

I. $y_1$, $y_2$ and $y_3$ are linearly independent on $-1 \leq x \leq 0$

II. $y_1$, $y_2$ and $y_3$ are linearly dependent on $0 \leq x \leq 1$

III. $y_1$, $y_2$ and $y_3$ are linearly independent on $0 \leq x \leq 1$

IV. $y_1$, $y_2$ and $y_3$ are linearly dependent on $-1 \leq x \leq 0$

Which one among the following is correct?

  1. Both I and II are true
  2. Both I and III are true
  3. Both II and IV are true
  4. Both III and IV are true

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Show answer and explanation

Correct answer: (B) Both I and III are true

Explanation

A nonzero polynomial of degree at most 2 has at most 2 roots, so $a + bx + cx^2 = 0$ on a whole interval forces $a = b = c = 0$. So $1$, $x$ and $x^2$ are independent on any interval of positive length, which makes I and III true.