Answer:
See picture and explanation below.
Step-by-step explanation:
With this information, the matrix A that you can find is the transformation matrix of T. The matrix A is useful because T(x)=Av for all v in the domain of T.
A is defined as [tex]T=([T(v_1)] [T(v_2] [T(v_3)])\text{ where }[T(v_i)][/tex] denotes the vector of coordinates of [tex]T(v_i)[/tex] respect to the basis [tex]<w_1,w_2,w_3>[/tex] (we can apply this definition because [tex]<v_1,v_2,v_3>[/tex] forms a basis for the domain of T).
The vector of coordinates can be computed in the following way: if [tex]T(v_i)=a_1w_1+a_2w_2+a_3w_3[/tex] then [tex][T(v_i)]=(a_1,a_2,a_3)^t[/tex].
Note that we have all the required information: [tex]T(v_1)=0w_1+1\cdot w_2+0w_3[/tex] then [tex][T(v_1)]=(0,1,0)^t[/tex]
[tex]T(v_2)=T(v_3)=1\cdot w_1+0w_2+0w_3[/tex] hence [tex][T(v_2)]=[T(v_3)]=(1,0,0)^t[/tex]
The matrix A is on the picture attached, with the multiplication A(1,1,1).
Finally, to obtain the output required at the end, use the properties of a linear transformation and the outputs given:
[tex]T(v_1+v_2+v_3)=T(v_1)+T(v_2)+T(v_3)=w_2+w_1+w_1=w_2+2w_1[/tex]
In this last case, we can either use the linearity of T or multiply by A.
