How to determine for the image room [closed]
Solution 1:
The images of the three standard basis vectors form a generating set for the image space, but they might be linearly dependent. What you need to do is solve a linear homogeneous system that finds any linear dependencies between those images; then any non-trivial dependency tells you that one of those image vectors can (and should) be removed, so as to obtain an independent generating set, i.e., a basis of the image space.