Linear Transformation PDF

Algebra Linear transformation Associate Professor Nguyen Trung Thanh Faculty of Data Science and Artificial Intelligence Functions Definition A function f : X → Y assigns to each element x of X (called domain) exactly one element y of Y (called codomain). A function is sometime called map or map...

Algebra Linear transformation Associate Professor Nguyen Trung Thanh Faculty of Data Science and Artificial Intelligence Functions Definition A function f : X → Y assigns to each element x of X (called domain) exactly one element y of Y (called codomain). A function is sometime called map or mapping. function NOT function 2/20 Functions Examples A univariate function: f : R → R with f(x) = x2 A bivariate function: f : R2 → R with f(x1 , x2 ) = 2x1 + 3x2 A multivariate function: f : Rn → R with f(x1 ,... , xn ) = a1 x1 + · · · + an xn A polynomial function: f : R2 → R with f(x1 , x2 ) = 2x21 + 3x22 − x1 x2 + 4x1 + 5x2 Definition One-to-one (or injective) function: f : X → Y: if x, x′ ∈ X, x ̸= x′ then f(x) ̸= f(x′ ) Onto function (or surjective) function: f : X → Y: for every y ∈ Y there exists x ∈ X such that f(x) = y Bijective function f : X → Y is the function that is both injective and surjective.3/20 Transformation Definition A transformation is a function T from Rn to Rm. Rn is called the domain of T Rm is called the codomain of T For x ∈ Rn , the vector T(x) is called the image of x under T The set of images {T(x) | x ∈ Rn } is called the range of T 4/20 Matrix Transformation Definition Let A = [aij ] ∈ Rm×n be a matrix. The matrix transformation associated to A is the transformation T from Rn to Rm , defined by T(x) = Ax. The range of T is the column space of A.         a11 x1 + · · · + a1n xn a11 a12 a1 n ..  ..  ..  ..  Ax = .  = x1 .  + x2 .  + · · · + xn .  am1 x1 + · · · + amn xn am1 am2 amn meaning the outputs of T(x) = Ax are exactly the linear combinations of the columns of A. 5/20 Projection onto the xy-plane      1 0 0 x x  0 1 0  y  =  y  0 0 0 z 0 6/20 Reflection [ ][ ] [ ] −1 0 x −x = 0 1 y y 7/20 Dilation [ ][ ] [ ] 1.5 0 x x = 1.5 0 1.5 y y 8/20 Rotation [ ][ ] [ ] 0 −1 x −y = 1 0 y x 9/20 Question Let T(x) = Ax is a matrix transformation from R2 to R3 where     1 1 [ ] 7 3 A =  0 1 and let u = , b = 5  4 1 1 7 Find T(u). Find a vector v ∈ R2 such that T(v) = b. Is there more than one? Does there exist a vector w ∈ R3 such that there is more than one v ∈ R2 with T(v) = w? Find a vector w ∈ R3 which is not in range of T. 10/20 One-to-one transformation Definition A transformation T from Rn to Rm is one-to-one if one of the following holds: for every b ∈ Rm , there is at most one vector x ∈ Rn such that T(x) = b for every x, y ∈ Rn with x ̸= y, then T(x) ̸= T(y). for every x, y ∈ Rn , if T(x) ̸= T(y) then x ̸= y. Examples The matrix transformation associated with the identify matrix is one-to-one. 11/20 One-to-one transformation Properties Let T : Rn → Rm be a matrix transformation associated with a matrix A. The following statements are equivalent: T is one-to-one for every b ∈ Rm , the equation T(x) = b has at most one solution for every b ∈ Rm , the equation Ax = b has a unique solution or is inconsistent Ax = 0 has only the trivial solution The columns of A are linearly independent The range of T has dimension n. 12/20 One-to-one transformation Examples Check which of the following matrix transformations are one-to-one?     1 0 1 0 1 [ ] [ ] 1 1 0 1 −1 2 A =  0 1 , B =  0 1 0 , C= , D= 0 1 1 −2 2 −4 1 0 0 0 0 13/20 Onto transformation Definition A transformation T from Rn to Rm is onto if one of the following holds: for every b ∈ Rm , there is at least one vector x ∈ Rn such that T(x) = b the range of T is equal to the codomain of T Examples The matrix transformation associated with identify matrix is onto. 14/20 Onto transformation Properties Let T : Rn → Rm be a matrix transformation associated with a matrix A. The following statements are equivalent: T is onto for every b ∈ Rm , the equation T(x) = b has at least one solution for every b ∈ Rm , the equation Ax = b is consistent The columns of A spans Rm The range of T has dimension m. 15/20 Onto transformation Examples Check which of the following matrix transformations are onto?   [ ] 1 0 [ ] 1 1 0 1 −1 2 A= , B =  0 1 , C= 0 1 1 −2 2 −4 1 0 16/20 Linear transformation Definition A transformation T from Rn to Rm is linear if T(x + y) = T(x) + T(y) T(rx) = rT(x) for all vectors x, y and all scalars r. Every matrix transformation is a linear transformation and vice versa. 17/20 Non-linear transformation Verify that following transformations are not linear? [ ] [ ] [ ] [ ] [ ] [ ] x |x| x xy x 2x + 1 T1 = , T2 = , T3 = = , y y y y y x − 2y 18/20 Composition of linear transformations Definition Let T : Rn → Rm and U : Rp → Rn be linear transformations. Their composition is the transformation T ◦ U : Rp → Rm defined by (T ◦ U)(x) = T(U(x)). 19/20 Composition of linear transformations Examples Let T : R3 → R2 and U : R2 → R3 be linear transformations:   [ ] 1 0 1 1 0 T(x) = x, and U(x) = 0 1 x. 0 1 1 1 0 The composition is a transformation T ◦ U : R2 → R2 , for which the associated matrix is   [ ] [ ] 1 0 1 1 1 1 0  = 0 1 1 1 0 1 1 1 0 20/20