Let T:R^{2}\rightarrow R^{2} such that T(v) = Av + b, where A is a 2\ri

Minerva Kline

Minerva Kline

Answered question

2021-11-08

Let T:R2R2 such that T(v) = Av + b, where A is a 22 matrix. (Such a transformation is called an affine transformation.) Prove that T is a linear transformation if and only if b = 0.

Answer & Explanation

Whouldess

Whouldess

Beginner2021-11-09Added 11 answers

Step 1
Given that T:R2r2 SUCH THAT T(v)=Av+b where A is 2 by 2 matrix.
Assume that T is linear transformation.
This implies that T(u+v)=T(u)+T(v) and T(αu)=aT(u).
Observe that,
T(u+v)=T(u)+T(v)
A(u+v)+b=Au+b+Av+b
Au+Av=Au+Av+b
Au+Av+b=Au+Av+2b
Au+Av=Au+Av+b
b=Au+Av-(Au+Av)
b=0
Step 2
Convtrsely,
assume thet b=0.This implies that T(v)=Av.
T(u+v)=A(u+v)
=Au+Av
T(u)+T(v)
Similarly,
T(αu)=A(αu)
=αAu
αT(u)
Thus, T is a linear transformation.

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