Question

state Rank-nullity theorem​

Answer 1

Answer:

hey mate here is ur ans

Step-by-step explanation:

The rank-nullity theorem states that the rank and the nullity (the dimension of the kernel) sum to the number of columns in a given matrix. If there is a matrix M with x rows and y columns over a field, then rank ( M ) + nullity ( M ) = y . \text{rank}(M) + \text{nullity}(M) = y. rank(M)+nullity(M)=y.

hope it help u

Answer 2

Vector space homomorphism or linear transformation :

Let U(F) and V(F) be two vector spaces over the same field F , then a mapping T : U → V is called a linear transformation or homomorphism of U into V if :

T(x + y) = T(x) + T(y) ∀ x , y ∈ U

T(ax) = aT(x) ∀ x ∈ U , a ∈ F

In another words , a mapping T : U(F) → V(F) is called a linear transformation or homomorphism of U into V if : T(ax + by) = aT(x) + bT(y) ∀ x , y ∈ U , a , b ∈ F .

Range and null space of linear transformation :

Let U(F) and V(F) be two vector spaces and let T be a linear transformation from U into V , then the image or the range of T is denoted by R(T) (or Im(T) or T(U)) is the set of all vectors y ∈ V such that T(x) = y , x ∈ U .

i.e. R(T) = {y ∈ V : T(x) = y , x ∈ U}

And the null space of T is denoted by N(T) is the set of all the vectors x ∈ U , such that T(x) = 0' where 0' is the zero element of V .

i.e. N(T) = {x ∈ U : T(x) = 0' ∈ V is the zero element}

But if we regard the linear transformation T from U to V as a vector space homomorphism of U into V , then the null space of T is also called the kernal of T and it will be denoted by Ker(T) .

♦ R(T) is a subspace of V and N(T) is a subspace of U .

Range and Nullity of a linear transformation :

Let T be a linear transformation from a vector space U(F) into a vector space V(F) , where U(F) is a finite dimensional vector space . Then the dimension of the range of T is called the rank of T and is denoted by ρ(T) ,

i.e. ρ(T) = rank of T = dim(R(T))

And the nullity of T is defined as the dimension of the null space of T and is denoted by η(T) ,

i.e. η(T) = nullity of T = dim(N(T))

Rank Nullity theorem :

If T : U → V is a linear transformation , where U is finite dimensional , then

Rank(T) + Nullity(T) = dim(U)

i.e. ρ(T) + η(T) = dim(U)