LinearTransformation.md (2873B)
1 # Linear Transformation 2 3 Khan 4 5 **Definition:** A linear transformation is a function with an input and output vector that respects addition and scalar multiplication. 6 7 ## Formally 8 9 The two rules where a and b are vectors and c is a scalar: 10 11 T(a + b) = T(a) + T(b) 12 13 T(ca) = cT(a) 14 15 This is necessary and sufficient for the function T to be a L.T. 16 17 This can be stated as the origin must remain fixed and all lines must remain lines. If we visualize this then the first requirement can be thought of as only allowing all grid lines to rotate but not offset. To visualize the second think that we can't have grid lines in the end that curved. As such, all uniform grid lines must be uniformly spaced in the end as well otherwise diagnol lines would become curved and thus the function would not respect lines. 18 19 ## Interesting Notes 20 21 When determining mappings of LTs we can use our knowledge of the multiplicative nature of LTs to say c <x, 0> = <cx,0> for any coeficcient c. This is powerful as it allows us to describe any (input) vector on the line that contains any other (input) vector that we already know the mapping for. 22 23 Ex. 24 25 [1] -> [5] 26 [0] [2] 27 28 Then we know 29 30 [2] -> [5x2] = [10] 31 [0] [2x2] [ 4] 32 33 Given this, if we know what the unit vectors map to we can then use them as a composite for all other mappings. In n dimensional space this means we have n vectors of length 1 where each vector has all zero components except one which is = 1. 34 35 When describing LTs in matrix form each column represents where a given unit vector will be mapped to. This is ordered so the first column will be the mapping of [1,..., 0] the second [0, 1, ..., 0] and so on. 36 37 **Important:** 38 39 Any LT can be represented as a matrix and all matrix multiplication is a LT. 40 41 ## Image 42 43 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. 44 45 The image of Z under T are all possible outputs of the function with inputs that are in Z. 46 47 ## Composition 48 49 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. 50 51 We call this composition the composition of T with S. 52 53 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. 54 55 If: 56 T(x) = Ax 57 S(x) = Bx 58 59 Then: 60 T(S(x)) = A(Bx) = ABx 61 62 Compositions of matricies are associative, but not commutative. 63 64 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.