Name:- Dhanraj Vaghela Branch:- Mechanical Sem:- 02 Enrollment From the desk of Dhanraj from SRICT

Isomorphism: Thm 6.9: (Isomorphic spaces and dimension) Pf: Two finite-dimensional vector space V and W are isomorphic if and only if they are of the same dimension. From the desk of Dhanraj from SRICT

It can be shown that this L.T. is both 1-1 and onto. Thus V and W are isomorphic. From the desk of Dhanraj from SRICT

One-to-one: one-to-onenot one-to-one From the desk of Dhanraj from SRICT

Some important theorems related to one to one transformation Thm 1: A linear transformation T : V -> W is one to one if and only if ker(T) ={0}. Thm 2: A linear transformation T : V -> W is one to one if and only if dim(ker(T)) = 0, i.e., nullity (T) = 0. Thm 3: A linear transformation T : V -> W is one to one if and only if rank(T)=dim V. Thm 4: If A is an m x n matrix and TA : R n -> R n is multiplication by A then T A is one to one if and only if rank (A)= n. Thm 5: If A is an n x n matrix and TA : R n -> R n is multiplication by A then T A is one to one if and only if A is an invertible matrix. From the desk of Dhanraj from SRICT

Onto: (T is onto W when W is equal to the range of T.) Thm 1: A linear transformation T : V -> W is onto if and only if rank (T) = dim W Thm 2: If A is an m x n matrix and T A : R n -> R m is multiplication by A then T A is onto if and only if rank (A) = m. Let T : V -> W be a linear transformation and let dim V = dim W (i) If T is one-to-one,then it is onto. (ii) If T is onto, then it is one-to-one. From the desk of Dhanraj from SRICT

Example : Sol: T:Rn→RmT:Rn→Rm dim(domain of T)rank(T)nullity(T)1-1onto (a)T:R3→R3(a)T:R3→R3 330Yes (b)T:R2→R3(b)T:R2→R3 220 No (c)T:R3→R2(c)T:R3→R2 321 Yes (d)T:R3→R3(d)T:R3→R3 321No From the desk of Dhanraj from SRICT

(Isomorphic vector spaces) The following vector spaces are isomorphic to each other. From the desk of Dhanraj from SRICT

Thank You From the desk of Dhanraj from SRICT