MATLAB PROJECT- VISUALIZING LINEAR TRANSFORMATIONS OF THE PLANE

MATLAB PROJECT- VISUALIZING LINEAR TRANSFORMATIONS OF THE PLANE

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MATLAB Project: Visualizing Linear Transformations of the Plane Name________________________ Purpose: To understand the standard matrix of a linear transformation. In particular, to see the geometric effect of how a 2x2 matrix transforms R 2 ; and conversely, to learn how to write 2x2 matrices that will transform R 2 in specific ways. Prerequisites : Section 1.9. A theorem from Section 1.4 is needed for the extra credit problem. MATLAB functions used : visdat and drawpoly from Laydata Toolbox Background . As shown in Theorem 10 in Section 1.9, when a linear transformation T: R n R n is given, it can be identified with a matrix, and this is an easy way to get a formula for the function. Let T: R n R n be a linear transformation and let e 1 , e 2 ,… , e n denote the columns of the nxn identity matrix. Figure out what each T( e i ) should be and write each T( e i ) as a column vector. If you then define the matrix A = [ T( e 1 )T ( e 2 ). .T ( e n )] ,thenit will be true that T( x )=A x for all x ,and Ax gives is a formula for the function. In other words, given a linear transformation T: R n R n , if you know its values at just the n independent vectors e 1 , e 2 ,. ., e n , then its value at every point x is determined! 1. Example . The 2x2 linear transformation that maps e 1 to e 1 + e 2 and e 2 to e 1 - e 2 is 1 1 1 1 . The 3x3 matrix transformation that maps e 1 to 1 2 3 , e 2 to 7 0 6 and e 3 to 1 4 5 is 1 7 1 4 0 2 5 6 3 . 2. Example . The function that reflects R 2 across the line y = -x is a linear transformation. Notice it must map e 1 to - e 2 and e 2 to - e 1 , so its matrix is 0 1 1 0 . See the sketch in Table 1 of Section 1.9. 3. Exercise . (hand) (a) Write a 2x2 matrix that maps e 1 to 4 e 2 and e 2 to - e 1 : (b) Write a 2x2 matrix that reflects R 2 across the line y = x : More background . A matrix transformation always maps a line onto a line or a point, and maps parallel lines onto parallel lines or onto points. (See exercises 25-28 in Section 1.8.) In the following question, you will verify these things for a particular matrix. 4. Exercise . (hand) Let M = 1 1 0 1 . (a) Explain why the function T( x )= M x maps the x-axis onto the line y = x, and why it maps the line y = 2 onto the line y = x+2. (Hints: A general point on the x-axis is of the form 0 t ; calculate M 0 t and interpret where those image points lie. Similarly, calculate M
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Page 2 of 6 MATLAB Project: Visualizing Linear Transformations of the Plane (b) Sketch here what you showed algebraically in 4(a). That is, sketch the x-axis and the line y = 2 on the
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MATLAB PROJECT- VISUALIZING LINEAR TRANSFORMATIONS OF THE PLANE

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