Final answer:
The image of an arbitrary vector v under a linear transformation T that is defined by matrix A is calculated by multiplying A by v, resulting in the vector Av.
Step-by-step explanation:
When a linear transformation T is applied to an arbitrary vector v, the image of v is found by multiplying the matrix A associated with T by the vector v: T(v) = Av. The resulting vector is the image of v under the transformation T. If vector v is represented in a Cartesian coordinate system as v = vxi + vyj + vzk, where vx, vy, and vz are the scalar components along the x, y, and z-axis respectively, and i, j, k are the unit vectors in those directions, then the image of v, Av, can be represented in the same system showing how the original vector has been transformed.