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2. We say that two matrices A and B commute if A B = B A. Matrices do not commute in general, but certain types of matrices do. Which of the following types of matrices commute? Assume matrices are 4 x 4. • scaling matrices S,S2 • translation matrices TT2 • rotation matrices, R,R2 • rotation and translation, RT • scaling and translation, ST scaling and rotation, SR To prove each one formally, you would need to write out a general form of each type of matrix and do the brute force calculation which wouldn't help you much in terms of how to think about it. Instead, try to visualize it. For cases of failure imagine a counterexample. 3. There are two ways to define a 3D dot product of two vectors u and v, namely the sum of U;V; for i = 1, ..3, and | u || v | cos(0). In lecture 2, I sketched out an argument for why these definitions are equivalent. Write out the same sketch for the 3D case.

2. We say that two matrices A and B commute if A B = B A. Matrices do not commute in general, but certain types of matrices do. Which of the following types of matrices commute? Assume matrices are 4 x 4. • scaling matrices S,S2 • translation matrices TT2 • rotation matrices, R,R2 • rotation and translation, RT • scaling and translation, ST scaling and rotation, SR To prove each one formally, you would need to write out a general form of each type of matrix and do the brute force calculation which wouldn't help you much in terms of how to think about it. Instead, try to visualize it. For cases of failure imagine a counterexample. 3. There are two ways to define a 3D dot product of two vectors u and v, namely the sum of U;V; for i = 1, ..3, and | u || v | cos(0). In lecture 2, I sketched out an argument for why these definitions are equivalent. Write out the same sketch for the 3D case.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.3: The Inverse Of A Matrix
Problem 44EQ
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2. We say that two matrices A and B commute if A B = B A. Matrices do not commute in general, but certain types of matrices do. Which of the following types of matrices commute? Assume matrices are 4 x 4. scaling matrices S,S2 translation matrices T1T2 • rotation matrices, R,R2 rotation and translation, RT scaling and translation, ST scaling and rotation, SR To prove each one formally, you would need to write out a general form of each type of matrix and do the brute force calculation which wouldn't help you much in terms of how to think about it. Instead, try to visualize it. For cases of failure imagine a counterexample. 3. There are two ways to define a 3D dot product of two vectors u and v, namely the sum of u;V; for i = 1,.3, and | u||v | cos(0). In lecture 2, I sketched out an argument for why these definitions are equivalent. Write out the same sketch for the 3D case.
Transcribed Image Text:2. We say that two matrices A and B commute if A B = B A. Matrices do not commute in general, but certain types of matrices do. Which of the following types of matrices commute? Assume matrices are 4 x 4. scaling matrices S,S2 translation matrices T1T2 • rotation matrices, R,R2 rotation and translation, RT scaling and translation, ST scaling and rotation, SR To prove each one formally, you would need to write out a general form of each type of matrix and do the brute force calculation which wouldn't help you much in terms of how to think about it. Instead, try to visualize it. For cases of failure imagine a counterexample. 3. There are two ways to define a 3D dot product of two vectors u and v, namely the sum of u;V; for i = 1,.3, and | u||v | cos(0). In lecture 2, I sketched out an argument for why these definitions are equivalent. Write out the same sketch for the 3D case.
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