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?

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 denotes the vector of coordinates of 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 then .

Note that we have all the required information: then

hence

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:

 

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

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