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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.

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  • $\begingroup$ Have you made any effort of your own to solve this? If so, please include those efforts in your question. $\endgroup$ – amd Dec 2 at 23:10
  • $\begingroup$ 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? $\endgroup$ – amd Dec 2 at 23:13
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Here are some hints in the form of relating the questions-as-asked to the concepts you describe.

(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?

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  • $\begingroup$ Thank you, it took me a while but I understood it. For my Kernel I obtained two basis, and for my image I obtained two basis. a) Basically the span of a basis in my kernel will be transformed to the point (0,0,0). b) I will use one of the two basis of my kernel plus any other linearly independent such that when It transforms I will have a line. C) Any the span of any basis (a line) linearly idependent from my kernel will become a line in R^3. D) I will have to use any two linearly idependent basis that are not in my kernel space thus they will turn to a plane in R^3. $\endgroup$ – Josue Dec 3 at 21:57

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