#CSIRNET #mathematics SEPT 2022 | Linear Algebra | Dimension of Vector Space-Fibonacci like sequence

This is Mahesh Yulekar and you are watching NextGen Maths. In this video we are going to discuss the problem of CSI and its September 2022 problem of linear algebra. Question is let us define a sequence a n, n belongs to n of real numbers to be a Fibonacci like sequence. That means a n equal to a n minus 1 plus a n minus 2 for n greater than equal to 3. What is the dimension of real vector space of Fibonacci like sequence? We have given four options. First option 1, second option 2, third option infinite and countable and the fourth option infinite and uncountable. So here we are going to discuss the real vector space of Fibonacci like sequence. So we construct SAT of Fibonacci like sequence V equal to a n such that a n equal to a n minus 1 plus n minus 2 for n greater than equal to 3. We define operation vector addition a n plus b n is equal to term wise addition a n plus b n for n belongs to capital N. Scalar multiplication k times a n equal to k a n for any scalar k. We can easily prove that the SAT V is a vector space with respect to the above operation. We are not going into detail but you can do by yourself. Now consider the following sequences p n equal to n belongs to N. First two terms is 1, 0 then the other remaining terms are the sum of previous two terms that is 1 plus 0 equal to 1 which will be the third term then 0 plus 1 equal to 1 which will be the fourth term then fifth term is 1 plus 1, 2 and continue. And similarly q n with the first two terms 0, 1 next term is 1, 2, 3, 5. These two sequences are independent with respect to the defined operation because one cannot be obtained by the linear combination of other. That means one cannot be obtained by scalar multiplication of the other. Therefore p n and q n are linearly independent with respect to the defined operation and moreover any other Fibonacci like sequence can be obtained using linear combination of p n and q n. Therefore these two sequences behave as a basis vectors for the vector space V which is a Fibonacci like sequences. Therefore B contain two elements p n and q n and B is behave as a basis for the vector space V. Basis B contain two elements therefore the dimension of V is 2 and which is a option number 2 and it is the correct answer. Thanks for watching. For more videos of CSI NET, please like, subscribe and press the bell icon of NextGenMaths.