Line Image Transformation with 3x3 Matrix - Vector Solution

[takercena]

Please help me solve this question:
Question: Find the images of the lines r = i - j + n(i + j + k) under transformation
M (matrix 3 x 3) =
(2 1 4)
(3 5 1)
(1 2 0)
in vector form.

 

[HallsofIvy]

The image under the transformation is just that matrix times the vector? But what is n? My first thought was that it was the parameter but then that is just a single line. If that is correct then r= i- j+ n(i+ j+ k)= (n+1)i+ (n-1)j+ nk and its image is
$$\left[\begin{array}{ccc}2 & 1 & 4 \\ 3 & 5 & 1 \\ 1 & 2 & 0\end{array}\right]\left[\begin{array}{c} n+1 \\ n-1 \\ n\end{array}\right]$$

 

[takercena]

Thank you. There is some confusion here. I am confuse when to use inverse matrix to find transformation. Can you explain this. Example : 1. Find the images under the transformation

 

[HallsofIvy]

I have no idea what you mean by that. You use the inverse matrix in order to find an "inverse image": x such that Mx= y. Given a matrix M, the image of x is, by definition Mx.