Introduction to Linear Transformations

1.8 Introduction to Linear Transformations Math 2331 – Linear Algebra 1.8 Introduction to Linear Transformations Math 2331, Linear Algebra 1 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation 1.8 Introduction to Linear Transformations Matrix Transformations Matrix Acting on Vector Matrix-Vector Multiplication Transformation: Domain and Range Examples Applications Computer Graphics Linear Transformation Definition Examples Matrix Transformations Math 2331, Linear Algebra 2 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Matrix Transformations Another Way to Vview Ax = b Matrix A is an object acting on x by multiplication to produce a new vector Ax or b. Example     2 −4   −8  3 −6  2 =  −12  3 1 −2 −4     2 −4   0  3 −6  2 =  0  1 1 −2 0 Math 2331, Linear Algebra 3 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Matrix Transformations Matrix Transformations Suppose A is m × n. Solving Ax = b amounts to finding all Rn which are transformed into vector b in Rm through multiplication by A. multiply by A in transformation “machine” Math 2331, Linear Algebra 4 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Transformation Transformation A transformation T from Rn to Rm is a rule that assigns to each vector x in Rn a vector T (x) in Rm . T : Rn −→ Rm Terminology Rn : domain of T Rm : codomain of T T (x) in Rm is the image of x under the transformation T Set of all images T (x) is the range of T Math 2331, Linear Algebra 5 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Matrix Transformations: Example Example  1 Let A =  2 0  Then if x =  0 1 . Define T : R2 −→ R3 by T (x) = Ax. 1      1 0   2 2 2 , T (x) = Ax =  2 1  = 5  1 1 0 1 1 Math 2331, Linear Algebra 6 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Matrix Transformations: Example Example  Let A =    2 1 −2 3 2 , u =  3 , b = and −5 10 −15 −10 1     3 c= . Define a transformation T : R3 → R2 by T (x) = Ax. 0 a. Find an x in R3 whose image under T is b. b. Is there more than one x under T whose image is b. (uniqueness problem) c. Determine if c is in the range of the transformation T . (existence problem) Solution: for x, or    x   1 −2 3  1  2 x2 = −5 10 −15 −10 x3 (a) Solve =  Math 2331, Linear Algebra 7 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Matrix Transformations: Example (cont.) Augmented matrix:     1 −2 3 2 1 −2 3 2 ∼ −5 10 −15 −10 0 0 0 0 x1 = 2x2 − 3x3 + 2 x2 is free x3 is free Let x2 = and x3 =  So x =  . Then x1 =  .  Math 2331, Linear Algebra 8 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Matrix Transformations: Example (cont.) (b) Is there an x for which T (x) = b? Free variables exist ⇓ There is more than one x for which T (x) = b (c) Is there an x for which T (x) = c? This is another way of asking if Ax = c is . Augmented matrix:     1 −2 3 3 1 −2 3 0 ∼ −5 10 −15 0 0 0 0 1 c is not in the of T . Math 2331, Linear Algebra 9 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Matrix Transformations: Applications Matrix transformations have many applications - including computer graphics Example   .5 0 . The transformation T : R2 → R2 defined by 0 .5 T (x) = Ax is an example of a contraction transformation. The transformation T (x) = Ax can be used to move a point x.        8 .5 0 8 4 u= T (u) = = 6 0 .5 6 3 Let A = Math 2331, Linear Algebra 10 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Matrix Transformations: Applications (cont.) Math 2331, Linear Algebra 11 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Linear Transformations If A is m × n, then the transformation T (x) = Ax has the following properties: T (u + v) = A (u + v) = = + + and T (cu) = A (cu) = Au = T (u) for all u,v in Rn and all scalars c. Linear Transformation A transformation T is linear if: 1 2 T (u + v) = T (u) +T (v) for all u, v in the domain of T . T (cu) =cT (u) for all u in the domain of T and all scalars c. Math 2331, Linear Algebra 12 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Linear Transformation: Definition Every matrix transformation is a linear transformation. RESULT If T is a linear transformation, then T (0) = 0 and T (cu + dv) =cT (u) +dT (v). Proof: T (0) = T (0u) = T (cu + dv) = T ( = T( T (u) = . )+T ( )+ T( Math 2331, Linear Algebra ) ) 13 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Matrix Transformations: Example Example      1 0 1 0 Let e1 = , e2 = , y1 =  0  and y2 =  1 . 0 1 2 1 2 3 Suppose T : R → R is a linear transformation which  maps e1   3 x1 into y1 and e2 into y2 . Find the images of and . 2 x2    Solution: First, note that T (e1 ) = and T (e2 ) = . Also  e1 + e2 = 3 2  Math 2331, Linear Algebra 14 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Matrix Transformations: Example (cont.) Then  T 3 2  =T( T (e1 ) + e1 + e2 ) = T (e2 ) = T (3e1 + 2e2 ) = 3T (e1 ) + 2T (e2 ) Math 2331, Linear Algebra 15 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Matrix Transformations: Example (cont.) Also  T x1 x2  =T( T (e1 ) + e1 + e2 ) = T (e2 ) = Math 2331, Linear Algebra 16 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Matrix Transformations: Example Example Define T : R3 → R2 such that T (x1 , x2 , x3 ) = (|x1 + x3 | , 2 + 5x2 ). Show that T is a not a linear transformation. Solution: Another way to write the transformation:     x1 |x1 + x3 |     x2 T = 2 + 5x2 x3 Provide a counterexample - example where T (0) = 0, T (cu) =cT (u) or T (u + v) = T (u) + T (v) is violated. A counterexample:     0     0 T (0) = T = 6= 0 which means that T is not linear. Math 2331, Linear Algebra 17 / 18 1.8 Introduction to Linear Transformations Matrix Transformation Example Linear Transformation Matrix Transformations: Example (cont.)   1 Another counterexample: Let c = −1 and u =  1 . Then 1       −1 |−1 + −1| 2     −1 T (cu) = T = = 2 + 5 (−1) −3 −1 and    1     1 cT (u) = −1T = −1 1 Therefore T (cu) 6= not   =  . T (u) and therefore T is . Math 2331, Linear Algebra 18 / 18