can someone please explain how to tell an image of a linear transformation

[obsidian canopy]

can someone please explain how to tell an image of a linear transformation
like I get how youre supposed to multiply a matrix with another matrix but how can u easily tell if the image is a reflection, contraction, etc?

[karmic frigate]

say A is a matrix representing your linear transformation.
Take a generic variable column vector X and look at the column vector AX.
Assume X is non-zero. Then ||AX||/||X|| being less than 1 for any X implies that you have a contraction

|| || denotes the euclidean norm

|| column[x y z] || = sqrt(x^2 + y^2 + z^2)

i think the converse also holds, but better check that

there's another way to look at this too, there're ways to define norms of matrices and then for the more usual ones, ||A|| < 1 implies that you have a contraction (and either the converse holds or you need to add one or two extra conditions, gotta check)

one norm on matrices is similar to the euclidean norm, aka you define
|| A || := sqrt( sum_{all i and j} |a_{i, j}|^2 )
where the a_{i, j} is the (i, j)-entry of A

for reflections, i think that if your transformation is on the euclidean plane, aka R^2 --> R^2, then
(x, y) |--> (-y, x) is a reflection iirc

reflection over the "y=x" line

but you can ask for a reflection over any given straight line, right

using analytic geometry you can determine how they are and see what you can use to identify them