Question

With bases V1, V2 , V3 andw1, w2 , W3, suppose T(v1) = w2 and T(v2) = T(v3) = w1 + w3 . T is a linear transformation. Find the matrix A and multiply by the vector (1, 1, 1). What is the output from T when the input is v1 + v2 + v3?

Answer 1

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 $T=([T(v_1)] [T(v_2] [T(v_3)])\text{ where }[T(v_i)]$ denotes the vector of coordinates of $T(v_i)$ respect to the basis (we can apply this definition because forms a basis for the domain of T).

The vector of coordinates can be computed in the following way: if $T(v_i)=a_1w_1+a_2w_2+a_3w_3$ then $[T(v_i)]=(a_1,a_2,a_3)^t$.

Note that we have all the required information: $T(v_1)=0w_1+1\cdot w_2+0w_3$ then $[T(v_1)]=(0,1,0)^t$

$T(v_2)=T(v_3)=1\cdot w_1+0w_2+0w_3$ hence $[T(v_2)]=[T(v_3)]=(1,0,0)^t$

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:

$T(v_1+v_2+v_3)=T(v_1)+T(v_2)+T(v_3)=w_2+w_1+w_1=w_2+2w_1$  

In this last case, we can either use the linearity of T or multiply by A.