1 Linear Equations in Linear Algebra 1.8 INTRODUCTION TO LINEAR TRANSFORMATIONS
LINEAR TRANSFORMATIONS A transformation (or function or mapping) T from R n to R m is a rule that assigns to each vector x in R n a vector T (x) in R m. The set R n is called domain of T, and R m is called the codomain of T. T : n The notation R R indicates that the domain of T is R n and the codomain is R m. For x in R n, the vector T (x) in R m is called the image of x (under the action of T ). The set of all images T (x) is called the range of T. See the figure on the next slide. m Slide 1.8-
MATRIX TRANSFORMATIONS For each x in R n, T (x) is computed as Ax, where A is an mn matrix. For simplicity, we denote such a matrix transformation by. x Ax The domain of T is R n when A has n columns and the codomain of T is R m when each column of A has m entries. Slide 1.8-3
MATRIX TRANSFORMATIONS The range of T is the set of all linear combinations of the columns of A, because each image T (x) is of the form Ax. Example 1: Let,,. and define a transformation R 3 R by, so that 1 3 A 3 5 1 7 u 1 3 c 5 T : T(x) Ax 1 3 x 3x 1 x1 T (x) Ax 3 5 3x 5x 1 x 1 7 x 7x 1. Slide 1.8-4
MATRIX TRANSFORMATIONS a. Find T (u), the image of u under the transformation T. b. Find an x in R whose image under T is b. c. Is there more than one x whose image under T is b? d. Determine if c is in the range of the transformation T. Slide 1.8-5
MATRIX TRANSFORMATIONS Solution: a. Compute b. Solve for x. That is, solve, or 1 3 5 T(u) Au 3 5 1 1 1 7 9 T(x) b Ax b 1 3 3 x1 3 5 x 1 7 5. ----(1). Slide 1.8-6
MATRIX TRANSFORMATIONS Row reduce the augmented matrix: 1 3 3 1 3 3 1 3 3 1 0 1.5 3 5 ~ 0 14 7 ~ 0 1.5 ~ 0 1.5 1 7 5 0 4 0 0 0 0 0 0 ----() x 1.5 x.5 1 1.5.5 Hence,, and. The image of this x under T is the given vector b. x Slide 1.8-7
MATRIX TRANSFORMATIONS c. Any x whose image under T is b must satisfy equation (1). From (), it is clear that equation (1) has a unique solution. So there is exactly one x whose image is b. d. The vector c is in the range of T if c is the image of some x in R, that is, if c T(x) for some x. This is another way of asking if the system is consistent. Ax c Slide 1.8-8
MATRIX TRANSFORMATIONS To find the answer, row reduce the augmented matrix. 1 3 3 1 3 3 1 3 3 1 3 3 3 5 0 14 7 0 1 0 1 ~ ~ ~ 1 7 5 0 4 8 0 14 7 0 0 35 The third equation, 0 35 system is inconsistent., shows that the So c is not in the range of T. Slide 1.8-9
SHEAR TRANSFORMATION A 1 3 0 1 T : T(x) Ax Example : Let R R defined by transformation.. The transformation is called a shear It can be shown that if T acts on each point in the square shown in the figure on the next slide, then the set of images forms the shaded parallelogram. Slide 1.8-10
SHEAR TRANSFORMATION The key idea is to show that T maps line segments onto line segments and then to check that the corners of the square map onto the vertices of the parallelogram. For instance, the image of the point T (u) 1 3 0 6 0 1, u 0 is Slide 1.8-11
LINEAR TRANSFORMATIONS 1 3 8 0 1 and the image of is. T deforms the square as if the top of the square were pushed to the right while the base is held fixed. Definition: A transformation (or mapping) T is linear if: i. T(u v) T(u) T(v) for all u, v in the domain of T; ii. T( cu) ct (u) for all scalars c and all u in the domain of T. Slide 1.8-1
LINEAR TRANSFORMATIONS Linear transformations preserve the operations of vector addition and scalar multiplication. Property (i) says that the result of first adding u and v in R n and then applying T is the same as first applying T to u and v and then adding T (u) and T (v) in R n. These two properties lead to the following useful facts. If T is a linear transformation, then T(0) 0 T(u v) ----(3) Slide 1.8-13
LINEAR TRANSFORMATIONS and. ----(4) for all vectors u, v in the domain of T and all scalars c, d. Property (3) follows from condition (ii) in the definition, because. Property (4) requires both (i) and (ii): If a transformation satisfies (4) for all u, v and c, d, it must be linear. (Set T( cu dv) ct (u) dt (v) T(0) T(0u) 0 T(u) 0 T( cu dv) T( cu) T( dv) ct (u) dt(v) c d 0 d 1 for preservation of addition, and set for preservation of scalar multiplication.) Slide 1.8-14
LINEAR TRANSFORMATIONS Repeated application of (4) produces a useful generalization: T( c v... c v ) ct(v )... c T(v ) 1 1 p p 1 1 p p ----(5) In engineering and physics, (5) is referred to as a superposition principle. Think of v 1,, v p as signals that go into a system and T (v 1 ),, T (v p ) as the responses of that system to the signals. Slide 1.8-15
LINEAR TRANSFORMATIONS The system satisfies the superposition principle if whenever an input is expressed as a linear combination of such signals, the system s response is the same linear combination of the responses to the individual signals. T : T(x) rx Given a scalar r, define R R by. T is called a contraction when dilation when r 1. 0r 1 and a Slide 1.8-16