Assume that T is a linear transformation. Find the standard matrix of T.T=RR^2rarrRR^4 such that T(e_1)=(7,1,7,1),and T(e_2)=(-8,5,0,0),where e_1=(1,0), and e_2=(0,1).

Burhan Hopper 2020-12-28 Answered

Assume that T is a linear transformation. Find the standard matrix of T.
T=R2R4 such that T(e1)=(7,1,7,1),.

andT(e2)=(8,5,0,0),where e1=(1,0),

ande2=(0,1)

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Expert Answer

gotovub
Answered 2020-12-29 Author has 98 answers

Let T:VW be a linear transformation. Suppose dim V=n,andS={v1,,vn} is an ordered basis for V and suppose dim W=mandB={w1,,wm} is an ordered basis for W
1.Calculate T(v1),T(v2),,T(vn)
2.Find the coordinate vectors (T(v1))B(T(v2))B,,(T(vm))B
3.Write the matrix with columns as the column vectors calculated in Step 2:
M=[(T(v1))B|(T(v2))B|(T(vm))B]
Clearly, here V=R2andW=R4.HereS={e1=(1,0),e2=(0,1)}is the standard ordered basis for VandB={ϵ1=(1,0,0,0),ϵ2=(0,1,0,0),ϵ3=(0,0,0,1),ϵ4=(0,0,0,1)} is the standard ordered basis for W.
Now we calculate [T(e1)]B,[T(e2)]B. Since T(e1)=(7,1,7,1) and T(e2)=(8,5,0,0) it follows that
T(e1)=7ϵ1+1ϵ2+7ϵ3+1ϵ4[T(e1)]B=(7,1,7,1)
T(e2)=8ϵ1+5ϵ2+0ϵ3+0ϵ4[T(e2)]B=(8,5,0,0)
Now we create the matrix M with columns as the column as the column vectors of [T(e1)]B,[T(e2)]B.
Therefore

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