This is a beautiful problem from ISI MStat 2016 Problem 4 PSB (sample) based on Vector space. It uses several concepts to solve it. We provide a detailed solution with prerequisites mentioned explicitly.
For each define a function
by
For every find the dimension of the null space of
.
Here we have to find the Kernel or null space of i.e {
:
} .
is defined as
So, , which gives
(i) if
(ii) if
(iii)
(iv)
(iii) & (iv) if
And if then
if c
.
Now for different values of c and using (i),(ii),(iii) and (iv) we will find the Null space as follows ,
=
Therefore for different values of c we will get different dimension of as follows ,
If c=-1 then . Hence the vectors {(1,0,0,0) , (0,1,1,0) ,(0,0,0,1) } spans
and they are Linearly Independent . Thus on this case dimension of null space
is 3 .
If c=0 then . Thus on this case dimension of null space
is 0.
If c=1 then .Hence the vectors { (0,1,-1,0) } spans
and they are Linearly Independent . Thus on this case dimension of null space
is 1 .
Finally if then
. Thus on this case dimension of null space
is 0.
This is a beautiful problem from ISI MStat 2016 Problem 4 PSB (sample) based on Vector space. It uses several concepts to solve it. We provide a detailed solution with prerequisites mentioned explicitly.
For each define a function
by
For every find the dimension of the null space of
.
Here we have to find the Kernel or null space of i.e {
:
} .
is defined as
So, , which gives
(i) if
(ii) if
(iii)
(iv)
(iii) & (iv) if
And if then
if c
.
Now for different values of c and using (i),(ii),(iii) and (iv) we will find the Null space as follows ,
=
Therefore for different values of c we will get different dimension of as follows ,
If c=-1 then . Hence the vectors {(1,0,0,0) , (0,1,1,0) ,(0,0,0,1) } spans
and they are Linearly Independent . Thus on this case dimension of null space
is 3 .
If c=0 then . Thus on this case dimension of null space
is 0.
If c=1 then .Hence the vectors { (0,1,-1,0) } spans
and they are Linearly Independent . Thus on this case dimension of null space
is 1 .
Finally if then
. Thus on this case dimension of null space
is 0.