commit e40014669d3f08c49c3b3dc60a9de707fe14a817 parent 5cdfa427e34e5d0a4a53487b746c82c99ac1a56b Author: Andrew <andrewlaack1@gmail.com> Date: Wed, 31 Jul 2024 21:22:04 -0500 Took some notes Diffstat:
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diff --git a/LinearTransformation.md b/LinearTransformation.md @@ -46,3 +46,20 @@ Any LT can be represented as a matrix and all matrix multiplication is a LT. The image of a linear transformation (im(T)) are all possible outputs of the function where the inupts of T are any vector in R^n. The image of Z under T are all possible outputs of the function with inputs that are in Z. + +## Composition + +The composition of linear transformations is T(S(x)) where S goes from R^n to R^m and T goes from R^m to R^l. The output of the interior L.T. must have the same codomain as the domain of the exterior L.T. + +We call this composition the composition of T with S. + +To construct the standard matrix of the composition we simply need to evaluate the output of the composition for each of the basis vectors that span the domain of S. This is true because the composition of two linear transformations is always a linear transformation as we know it is still additive and scalar multiplicative. + +If: +T(x) = Ax +S(x) = Bx + +Then: +T(S(x)) = A(Bx) + +With this definition it is intuitive that the standard matrix of the composition is A times B where A and B are the standard matricies of the L.T.s T and S.