Sept. 3, 2013 Math 3312 sec 003 Fall 2013

Transcription

1 Sept. 3, 2013 Math 3312 sec 003 Fall 2013 Section 1.8: Intro to Linear Transformations Recall that the product Ax is a linear combination of the columns of A turns out to be a vector. If the columns of A are vectors in R m, and there are n of them, then A is an m n matrix, the product Ax is defined for x in R n, and the vector b = Ax is a vector in R m. So we can think of A as an object that acts on vectors x in R n (via the product Ax) to produce vectors b in R m. September 3, / 30

2 Transformation from R n to R m Definition: A transformation T (a.k.a. function or mapping) 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. Some relevant terms and notation include R n is the domain and R m is called the codomain. For x in the domain, T (x) is called the image of x under T. The collection of all images is called the range. The notation T : R n R m may be used to indicate that R n is the domain and R m is the codomain. If T (x) is defined by multiplication by the m n matrix A, we may denote this by x Ax. September 3, / 30

3 Matrix Transformation Example Let A = mapping T (x) = Ax. (a) Find the image of the vector u =. Define the transformation T : R 2 R 3 by the [ 1 3 ] under T. September 3, / 30

4 A = (b) Determine a vector x in R 2 whose image under T is September 3, / 30

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6 (c) Determine if A = is in the range of T. September 3, / 30

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8 Linear Transformations Definition: A transformation T is linear provided (i) T (u + v) = T (u) + T (v) for every u, v in the domain of T, and (ii) T (cu) = ct (u) for every scalar c and vector u in the domain of T. Every matrix transformation (e.g. x Ax) is a linear transformation. And it turns out that every linear transformation from R n to R m can be expressed in terms of matrix multiplication. September 3, / 30

9 A Theorem About Linear Transformations: If T is a linear transformation, then T (0) = 0, T (cu + dv) = ct (u) + dt (v) for scalars c, d and vectors u, v. And in fact T (c 1 u 1 + c 2 u c k u k ) = c 1 T (u 1 ) + c 2 T (u 2 ) + + c k T (u k ). September 3, / 30

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11 Example Let r be a nonzero scalar. The transformation T : R 2 R 2 defined by T (x) = rx is a linear transformation 1. Show that T is a linear transformation. 1 It s called a contraction if 0 < r < 1 and a dilation when r > 1 September 3, / 30

12 Figure : Geometry of dilation x 2x. The 4 by 4 square maps to an 8 by 8 square. September 3, / 30

13 Section 1.9: The Matrix for a Linear Transformation Elementary Vectors: We ll use the notation e i to denote the vector in R n having a 1 in the i th position and zero everywhere else. e.g. in R 2 the elementary vectors are e 1 = [ 1 0 in R 3 they would be 1 e 1 = 0, e 2 = 0 and so forth. ] [ 0, and e 2 = ],, and e 3 = Note that in R n, the elementary vectors are the columns of the identity I n September 3, / 30

14 Matrix of Linear Transformation Let T : R 2 R 4 be a linear transformation, and suppose 0 1 T (e 1 ) = 1 2, and T (e 2) = Use the fact that T is linear, and the fact that for each x in R 2 we have [ ] [ ] [ ] x1 1 0 x = = x 1 + x 0 2 = x 1 2 e 1 + x 2 e 2 x 2 to find a matrix A such that T (x) = Ax for every x R 2. September 3, / 30

15 T (e 1 ) = , and T (e 2) = September 3, / 30

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17 Theorem Let T : R n R m be a linear transformation. There exists a unique m n matrix A such that T (x) = Ax for every x R n. Moreover, the j th column of the matrix A is the vector T (e j ), where e j is the j th column of the n n identity matrix I n. That is, A = [T (e 1 ) T (e 2 ) T (e n )]. The matrix A is called the standard matrix for the linear transformation T. September 3, / 30

18 Example Let T : R 2 R 2 be the scaling trasformation (contraction or dilation for r > 0) defined by Find the standard matrix for T. T (x) = rx, for positive scalar r. September 3, / 30

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20 Example Let T : R 2 R 2 be the rotation trasformation that rotates each point in R 2 counter clockwise about the origin through an angle φ. Find the standard matrix for T. September 3, / 30

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22 Example 2 Let T : R 2 R 2 be the projection tranformation that projects each point onto the x 1 axis T ([ x1 Find the standard matrix for T. x 2 ]) = [ x1 0 ]. 2 See pages in Lay for matrices associated with other geometric tranformation on R 2 September 3, / 30

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24 One to One, Onto Definition: A mapping T : R n R m is said to be onto R m if each b in R m is the image of at least one x in R n i.e. if the range of T is all of the codomain. Definition: A mapping T : R n R m is said to be one to one if each b in R m is the image of at most one x in R n. September 3, / 30

25 Determine if the transformation is one to one, onto, neither or both. T (x) = [ ] x. September 3, / 30

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27 Some Theorems Theorem (11): Let T : R n R m be a linear transformation. Then T is one to one if and only if the homogeneous equation T (x) = 0 has only the trivial solution. September 3, / 30

28 Some Theorems Theorem (12): Let T : R n R m be a linear transformation, and let A be the standard matrix for T. Then (i) T is onto if and only if the columns of A span R m, and (ii) T is one to one if and only if the columns of A are linearly independent. September 3, / 30

29 Example Let T (x 1, x 2 ) = (x 1, 2x 1 x 2, 3x 2 ). Verify that T is one to one. Is T onto? September 3, / 30

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