# Thread: Linear Map

1. ## Linear Map

Hello,

(a) Does there exist a linear map T : R^11-------->R^5 such that rank(T)=null(T)?

My attempt:Never because rank(T)+null(T)=dim(R^11)=11 , we obtain 2rank(T)=11, contradicting the fact that rank(T) is an integer.

(b) Does there exist a linear map T : R^6-------->R^2 such that 2rank(T) = nul(T)?

Yes. rank(T)+null(T)=dimm(R^6)=6
3rank(T)=6 so rank(T)=2

are these correct?

Thanks

2. Originally Posted by charikaar
Hello,

(a) Does there exist a linear map T : R^11-------->R^5 such that rank(T)=null(T)?

My attempt:Never because rank(T)+null(T)=dim(R^11)=11 , we obtain 2rank(T)=11, contradicting the fact that rank(T) is an integer.

(b) Does there exist a linear map T : R^6-------->R^2 such that 2rank(T) = nul(T)?

Yes. rank(T)+null(T)=dimm(R^6)=6
3rank(T)=6 so rank(T)=2

are these correct?

Thanks
Basically yes, but I'd require from a student to produce a specific example in (b), and not only to show that it is possible.

Tonio

3. Originally Posted by tonio
Basically yes, but I'd require from a student to produce a specific example in (b), and not only to show that it is possible

Tonio
can you help me with example please? thanks

4. Originally Posted by charikaar
can you help me with example please? thanks

Any onto map $\mathbb{R}^6 \rightarrow \mathbb{R}^2$ will do. I bet you can come up with at least 2 very simple such maps.

Tonio

5. is this correct

Thanks

6. Originally Posted by charikaar
is this correct

Thanks

No, of course i'ts not right. You have to produce a map from $\mathbb{R}^6\,\,to\,\,\mathbb{R}^2$, and your map above is defined only on two lin. indep. vectors in $\mathbb{R}^6$ and besides this it maps to... $\mathbb{R}^6$ again!

Tonio