Question

Let U(F) and V(F) be the two vector spaces over the field F and T in linear transformation

from U into V. U is finite dimensional then.

Rank (T) + Nullity (T) = dim U.​

Answer 1

Step-by-step explanation:

u(f) & v(f) are vector spaces

T:U->v is a linear transformation

U is finite dimensional

By know theorem we have

N(T) is a subspaces of U(f)

=>N(T) is also finite dimensional

such that S1={

alpha1,------alpha2}is a basis of N(T)

=>dim N(T)=k

and T(alpha1)=?

Answer 2

Relation between rank, nullity and dimension in linear mappings:

Given:

Let U(F) and V(F) be the two vector spaces over the field F and T in linear transformation.

⇒ T: V→ U

To Find:

Rank (T) + Nullity (T) = dim U.​

Solution:

• The rank of T is defined as the dimension of its image.
• The nullity of F is defined to be the dimension of its kernel.
• ∴ rank (T) = dim (Image F) and Nullity (T) = dim (Ker F)

⇒ dim (U) = dim (Image F) + dim (Ker F)

⇒ dim (U) = rank (T) +Nullity (T)

Hence the required condition i.e., Rank (T) + Nullity (T) = dim U is proved