Question

In: Advanced Math

If T1 , T2 : Rn → Rm are linear maps with range(T1) = range(T2), show...

If T1 , T2 : Rn → Rm are linear maps with range(T1) = range(T2), show that there exists an invertible linear map U:Rn →Rn so that T1=U◦T2

Expert Solution

be two linear transformation with range (T₁) = range(T₂) .

Let { v₁ , v₂ ,...,v_m } be a basis of range(T₁) .

There exist { u₁ , u₂ , ... ,u_m} and{ w₁ , w₂ , ... ,w_m} such that

T₁( u₁ ) = v₁ , T₂( w₁) = v₁

T₁( u₂) = v₂ , T₂ ( w₂ ) = v₂

_{..................}

T_m( u_m ) = v_m , T_m( w_m ) = v_m .

Since { v₁ , v₂ ,...,v_m } be a basis of range(T₁) so { u₁ , u₂ , ... ,u_m} and { w₁ , w₂ , ... ,w_m} are also linearly independent because image of linearly dependent set is linearly dependent .

Now since { u₁ , u₂ , ... ,u_m} and { w₁ , w₂ , ... ,w_m} are linearly independent so they can be extend to a basis of .

Let { u₁ , u₂ , ... ,u_{m ,} u_m₊₁ ,..., u_n } and { w₁ , w₂ , ... ,w_m , w_m+1 , ...,w_n } are basis of .

Define , is defined by ,

U(u_i ) = w_i for all i = 1,2 ,...,n

Then

Colby Messinger answered 2 years ago

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