GATE 2016 | MATHS | Q-11

Question

Let \( \mathbf{v}, \mathbf{w}, \mathbf{u} \) be a basis of \( \mathbb{V} \). Consider the following statements P and Q:
(P) : \( \{\mathbf{v} + \mathbf{w}, \mathbf{w} + \mathbf{u}, \mathbf{v} - \mathbf{u}\} \) is a basis of \( \mathbb{V} \).
(Q) : \( \{\mathbf{v} + \mathbf{w} + \mathbf{u}, \mathbf{v} + 2\mathbf{w} - \mathbf{u}, \mathbf{v} - 3\mathbf{u}\} \) is a basis of \( \mathbb{V} \).

Which of the above statements hold TRUE?
(A) both P and Q (B) only P
(C) only Q (D) Neither P nor Q

Answer 1

condition for basis

1. vectors need to be linearly independent
2. should cover the entire space of V

as $\{u,v,w\}$ form the basis so they must be Linearly independent

1.  $v-u = (v+w)-(w+u)$ as the third vector can be represented as combination of other two . It cannot form the basis for V as they are linearly dependent vectors. So P doesn’t form basis of V
2. notice that third vector doesn’t contain w , so lets try to eliminate w , $2k(v+w+u) – k(v+2w-u) = kv + 3ku$ as notice that there is no k which can make it equal to $v-3u$ , hence we can conclude all vectors here are linearly independent and there are 3 linearly independent vectors so the rank must also be 3 which means they cover the entire space of V . so Q forms basis of V
3. final ans –> C – only Q