Final answer:
To find out if the vector y is in the range of the linear mapping L, we need to set up and solve a system of linear equations from L(x) = y. If a solution exists, it provides the vector x for which L(x) = y. If no solution exists, y is not in the range of L.
Step-by-step explanation:
The student's question concerns determining whether a vector y lies in the range of a specific linear mapping L and, if it does, finding a corresponding vector x in the domain. To solve this, we need to set up and solve a system of linear equations derived from the equation L(x) = y. The components of the vector y and the mapping L provide the equations for the system. If a solution exists, it suggests that y is in the range of L, and the solution gives the corresponding vector x in V such that L(x) = y.
To determine if y is in the range of L, we compare the components of y with the image of L applied to a general vector x in the domain V. This process involves equating the components of y to the corresponding components in the image of L(x), thereby setting up a system of linear equations. If we can find values for x¹ and x² that satisfy all equations, then y is indeed in the range of L. Moreover, the found values of x¹ and x² give us the vector x such that L(x) = y. If no solution exists, then y is not in the range of L.