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