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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 .

- kernel or Null space of a linear transformation
- Dimension
- Spanning & Linearly Independent vectors of a vector space

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 .

- kernel or Null space of a linear transformation
- Dimension
- Spanning & Linearly Independent vectors of a vector space

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.

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