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Let T : R? → R? be the linear transformation which reflects across the line y = -2x and then ()- -3x- 4y –4x + 3y stretches in the x and y directions, each by a factor of 5. That is, T 3.1 Verify, using what we've learned in class, that the formula given for T indeed performs the transformation described geometrically above (reflecting across y = -2x and stretching by a factor of 5.) 3.2 Sketch a single copy of R2 which contains the following: T(e1) and T(e2), • The fundamental parallelogram for T (i.e. the image of the unit square under T), • The line y = -2x. 3.3 Explain why your drawing from the previous part implies that neither e nor ez is an eigenvector for AT. 3.4 Determine two basic eigenvectors vị and v2 for AT (you do not need to determine the

Let T : R? → R? be the linear transformation which reflects across the line y = -2x and then ()- -3x- 4y –4x + 3y stretches in the x and y directions, each by a factor of 5. That is, T 3.1 Verify, using what we've learned in class, that the formula given for T indeed performs the transformation described geometrically above (reflecting across y = -2x and stretching by a factor of 5.) 3.2 Sketch a single copy of R2 which contains the following: T(e1) and T(e2), • The fundamental parallelogram for T (i.e. the image of the unit square under T), • The line y = -2x. 3.3 Explain why your drawing from the previous part implies that neither e nor ez is an eigenvector for AT. 3.4 Determine two basic eigenvectors vị and v2 for AT (you do not need to determine the

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3 Let T : R? → R? be the linear transformation which reflects across the line y = -2x and then -3x-4y %3D -4x +3y stretches in the x and y directions, each by a factor of 5. That is, T 3.1 Verify, using what we've learned in class, that the formula given for T indeed performs the transformation described geometrically above (reflecting across y = -2x and stretching by a factor of 5.) 3.2 Sketch a single copy of R2 which contains the following: • T(e1) and T(e2), • The fundamental parallelogram for T (i.e. the image of the unit square under T), • The line y = -2x. 3.3 Explain why your drawing from the previous part implies that neither e nor ez is an eigenvector for AT. 3.4 Determine two basic eigenvectors vị and v2 for AT (you do not need to determine the eigenvalue(s) associated to those basic eigenvectors). Justify your answer using only a geometric explanation explicitly relying on and referring to your drawing from the previous part, or to a new drawing, if you wish. Note: Algebraic work will not be marked for parts 3.3 and 3.4 of this question. For instance, computations involving AT will be ignored. Instead you need to justify your answers purely using picture(s) and explanations relying on those picture(s).
Transcribed Image Text:3 Let T : R? → R? be the linear transformation which reflects across the line y = -2x and then -3x-4y %3D -4x +3y stretches in the x and y directions, each by a factor of 5. That is, T 3.1 Verify, using what we've learned in class, that the formula given for T indeed performs the transformation described geometrically above (reflecting across y = -2x and stretching by a factor of 5.) 3.2 Sketch a single copy of R2 which contains the following: • T(e1) and T(e2), • The fundamental parallelogram for T (i.e. the image of the unit square under T), • The line y = -2x. 3.3 Explain why your drawing from the previous part implies that neither e nor ez is an eigenvector for AT. 3.4 Determine two basic eigenvectors vị and v2 for AT (you do not need to determine the eigenvalue(s) associated to those basic eigenvectors). Justify your answer using only a geometric explanation explicitly relying on and referring to your drawing from the previous part, or to a new drawing, if you wish. Note: Algebraic work will not be marked for parts 3.3 and 3.4 of this question. For instance, computations involving AT will be ignored. Instead you need to justify your answers purely using picture(s) and explanations relying on those picture(s).
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