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.
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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Please solve both parts as soon as possible
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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