# Linear Transformations (Input and output space)

I am stuck with the following problem. I asked for help in Chegg but I am rather unsatisfied with their answer it feels as though they simply cherry picked points that satisfy the conditions. I am wondering whether there is a way to solve this using the concepts of Kernel(T) and Image(T) if so can I get a suggestion on how to tackle it?

Consider the linear transformation T:$$R^4$$$$R^3$$, T(x,y,z,t)=(x−y−z,x+y+z,3x).

(a) If possible, find a line in the input space that is T-transformed into a point of the output space.

(b) If possible, find a plane in the input space that is T-transformed into a line of the output space.

(c) If possible, find a line in the input space that is T-transformed into a line of the output space.

(d) If possible, find a plane in the input space that is T-transformed into a plane of the output space.

• Have you made any effort of your own to solve this? If so, please include those efforts in your question. – amd Dec 2 at 23:10
• Are these lines and planes assumed to be subspaces of the corresponding spaces or can they be any lines/planes? I.e., must they pass through the origin? – amd Dec 2 at 23:13

(a) If you can find a nonzero vector $$x$$ in the kernel of $$T$$, then the line spanned by $$x$$ has this property.
(b) If you can find two linearly independent vectors $$x$$ and $$y$$ such that $$x$$ is in the kernel of $$T$$ but $$y$$ is not, then the plane spanned by $$x$$ and $$y$$ has this property.
(c) If you can find a nonzero vector $$x$$ that is not in the kernel of $$T$$, then the line spanned by $$x$$ has this property.
(d) It's not quite enough to find two linearly independent vectors $$x$$ and $$y$$ neither of which is in the kernel of $$T$$. Do you see why? What stronger property would need to be satisfied?