Final answer:
The question asks for details on constructing the transformation matrix of a linear transformation given the images of two vectors. By placing the resultant vectors in as columns of the matrix, we can partially construct this matrix.
Step-by-step explanation:
The question deals with a linear transformation that is applied to two different vectors, yielding two resultant vectors. The information given allows us to determine the transformation matrix that defines this linear transformation.
When we are given the transformation of the basis vectors, it is possible to construct the transformation matrix by using the images of these basis vectors as the columns of the matrix. Here, if T is our transformation matrix, then T applied to the first vector [1, 2, 3] gives [1, 7, -1], and T applied to the second vector [1, 1, 0] gives [-2, 5, 3]. These two outcomes give us the first two columns of the transformation matrix. Since we don't have the image of the third basis vector, we can only partially construct the transformation matrix.
The process involves setting up a system of linear equations based on the given transformations, and solving for the unknown entries in the transformation matrix. In general, with enough information, this matrix can be fully determined and used to transform any vector from the pre-image space to the image space.