CSIR NET MATHEMATICS | Linear Algebra | One to One, Onto Map Q.72(A) Q.63(B) Q.69(C)

Welcome friends. In this video we are going to discuss the problem of Linear Algebra from CSINet June 2019. Example number 69 from booklet C marks 4.75. Let V be a finite dimensional vector space over R and T be a linear map. Can you always write T equal to T2 o T1 composite map from some linear map T1V2W and T2W2V where W is some finite dimensional vector space and such that first option both T1 and T2 are onto second option both T1 and T2 are 1 to 1 third option T1 onto T2 1 to 1 and the fourth option T1 1 to 1 T2 onto now it's a problem of decomposition of linear map T = T2 o T1 and there a different four option for T1 and T2 note that V is a finite dimensional vector space now option one suppose option one is correct that T can be decomposed into T2 o T1 where T1 and T2 both are onto map and this is true for all T we have result for finite dimensional vector spaces composition of onto linear map is onto therefore T1 and T2 are onto map therefore its composition T2 o T1 equal to T is onto and it is always true because we assume that T can be decomposed into T2 o T1 for all T but we are able to find some linear transformation which are not onto just for example orthogonal projection T defined from R2 to R2 T(x,y) equal to (x,0) now every element of V in co-domain has at least one u in domain such that tu equal to v then the map is called onto map this is a definition of onto map vector 0 1 from r2 co-domain space we cannot associate any vector u such that tu equal to v because the second component is always 0 in the image part and therefore the definition says that for every v there should be u in domain space here which is not possible therefore orthogonal projection is not an onto map and therefore the first option that both t1 and t2 are onto is wrong now option number two suppose option two is correct and which is both t1 and t2 are one to one for finite dimensional vector spaces composition of one-to-one linear map is one-to-one therefore t1 is one-to-one t2 is one-to-one therefore t is also one-to-one map and it is true for all t therefore always t is one-to-one map counter example the same example orthogonal projection txy equal to x0 from r2 to r2 it is not an one-to-one map so one-to-one map means what tu equal to tv then u equal to v that means image of two vector is same then vector must be same here we have two vectors u equal to 1 1 and v equal to 1 2 then their images are equal that means vector is not equal but that images are equal or in other words images are equal but the vectors may be different therefore this map is not one to one map so the second option this says that T is always one to one which is also wrong option number 3 where we required T1 is onto map and T2 is one-to-one map so we restrict W equal to TV TV is a range space of V now we define T1 from V to W and it is T of V equal to TV that means image of T1 is nothing but the image of so we can easily prove that this is an onto map now the T2 we require one-to-one map and the simplest one-to-one map is identity map and T2 V equal to V now if we take the composition of T2 T1 T2 o T1 equal to T2 T1 V and image of T1 V is nothing but the TV and T2 is identity map so we will get the TV therefore T equal to T2 o T1 so we can decompose T into the composite map T2 o T1 when T1 is onto and T2 is one to one and it is true for all the linear transformation therefore option number three is correct now once you are able to prove the option number three we can easily prove option number four if you remember this result result is for finite dimensional vector spaces V and W the linear map TV to W is one to one map if and only if it is onto so it is one one then it is onto map so the same map which defined for the option 3 we will work here W is equal to TV T1 V to W is equal to t1 v equal to tv it is one to one map because it is on to from this result and t2 w to v equal to t2 v equal to v it is on to map because it is a one one from this result and the same map will work and we'll get the same result therefore t equal to t2o t1 and therefore option number four is correct thanks for watching for more videos like share subscribe please the bell icon of NextGen Plus