Question

Let V be the set of all vectors of the form [x, y, z] with x, y, and z real. Find a matrix A such that A[x, y, z] = [2x + 3y - z, x - 2y + 4z].

Answer 1

The matrix A that transforms the three-dimensional vector [x, y, z] into the two-dimensional vector [2x + 3y - z, x - 2y + 4z] is given by A = [[2, 3, -1],[1, -2, 4]].

Step-by-step explanation:

The student is asking to find a matrix A that transforms a three-dimensional vector [x, y, z] into a two-dimensional vector [2x + 3y - z, x - 2y + 4z]. This transformation can be represented by a matrix multiplication where matrix A when multiplied with the three-dimensional vector yields the desired two-dimensional result. Such a matrix A can be found by arranging the coefficients of the variables x, y, and z in the rows corresponding to the resulting vectors' components. Thus, the matrix A is:

\[A = \begin{bmatrix}2 & 3 & -1\\1 & -2 & 4\end{bmatrix}\]

Each row of matrix A corresponds to the coefficients of x, y, and z for each component of the transformed vector.