- #1

bonildo

- 14

- 1

**Summary::**linear transformations

Hello everyone, firstly sorry about my English, I'm from Brazil.

Secondly I want to ask you some help in solving a question about linear transformations.

Here is the question:Consider the linear transformation described by the matrix [itex] \mathsf{A} \in \Re ^{2x2}

given by: [/itex]

[itex] A =

\begin{pmatrix}

1 & 1 \\

-1 & 1 \\

\end{pmatrix}

[/itex]

a) Find the representation of the linear transformation in the basis V={v1,v2}, where v1=transpose(1,1) , v2=transpose(2,0)

My approach:

Choosing and arbitrary vector in the vector space that V span then it can be write as a linear combination of the basis:

[itex] v=(x,y)=a1(1,1)+a2(2,0) [/itex]

Applying T on both sides:

[itex] T(v)=T((x,y))=a1T(1,1)+a2T(2,0) [/itex]

Finding T(1,1) and T(2,0):

[itex] T(1,1)=A*(1,1) =(2,0) [/itex]

[itex] T(2,0)=A*(2,0) = (2,-2) [/itex]

then:

[itex] T((x,y))= (2 a1 + 2 a2, -2 a2) [/itex]

Solving for a1 and a2:

[itex] a1=(x+y)/2 [/itex]

[itex] a2=-y [/itex]

and finally T(x,y):

[itex] T(x,y)=(x+y)/2 (2,0) +(-y)(2,-2) = (x-y,2y) [/itex]But when I substitute T(x,y) with (1,1) I don't get the same answer as A*(1,1) . Can someone help me with it ?

T(1,1)=(1-1,2*1) =(0,2)

and

A*(1,1) = (2,0)

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