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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 !!

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.

Linear Transformation

Null Space

Dimension

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

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

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 !!

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.

Linear Transformation

Null Space

Dimension

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

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

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