Get inspired by the success stories of our students in IIT JAM MS, ISI  MStat, CMI MSc DS.  Learn More

# ISI MStat PSB 2014 Problem 1 | Vector Space & Linear Transformation This is a very beautiful sample problem from ISI MStat PSB 2014 Problem 1 based on Vector space and Eigen values and Eigen vectors . Let's give it a try !!

## Problem- ISI MStat PSB 2014 Problem 1

Let where n is an odd positive integer. Let be
the vector space of all functions from E to , where the vector space
operations are given by , for for (a) Find the dimension of (b) Let be the map given by Show that T is linear.
(c) Find the dimension of the null space of T.

### Prerequisites

Linear Transformation

Null Space

Dimension

## Solution :

While doing this problem we will use a standard notation for vectors of canonical basis i..e . In they are and .

(a) For and , let be the function in which maps and where and . Then is a basis of .

It looks somewhat like this ,  , Hence , dimension of is 3n.

(b) To show T is linear we have to show that for some scalar a,b . .

Hence proved .

(c) gives so, the values of for the last points are opposite to first so we can freely assign the values of f for first to any of .Hence, the null space has dimension ## Food For Thought

let be a non singular linear transformation.Prove that there exists a line passing through the origin that is being mapped to itself.

Prerequisites : eigen values & vectors and Polynomials

## Subscribe to Cheenta at Youtube

This is a very beautiful sample problem from ISI MStat PSB 2014 Problem 1 based on Vector space and Eigen values and Eigen vectors . Let's give it a try !!

## Problem- ISI MStat PSB 2014 Problem 1

Let where n is an odd positive integer. Let be
the vector space of all functions from E to , where the vector space
operations are given by , for for (a) Find the dimension of (b) Let be the map given by Show that T is linear.
(c) Find the dimension of the null space of T.

### Prerequisites

Linear Transformation

Null Space

Dimension

## Solution :

While doing this problem we will use a standard notation for vectors of canonical basis i..e . In they are and .

(a) For and , let be the function in which maps and where and . Then is a basis of .

It looks somewhat like this ,  , Hence , dimension of is 3n.

(b) To show T is linear we have to show that for some scalar a,b . .

Hence proved .

(c) gives so, the values of for the last points are opposite to first so we can freely assign the values of f for first to any of .Hence, the null space has dimension ## Food For Thought

let be a non singular linear transformation.Prove that there exists a line passing through the origin that is being mapped to itself.

Prerequisites : eigen values & vectors and Polynomials

## Subscribe to Cheenta at Youtube

This site uses Akismet to reduce spam. Learn how your comment data is processed.

### Knowledge Partner  