Math 260, Spring 2017, Assignment 6
From cartan.math.umb.edu
I tell them that if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh.
- - Thomas Mann, The Magic Mountain
Carefully define the following terms, then give one example and one non-example of each:[edit]
- Orthogonal projection (of a vector on a line).
- Composition (of two transformations).
- Inverse (of a transformation).
Carefully state the following theorems (you do not need to prove them):[edit]
- Theorem concerning the linearity of the composition of linear transformations.
- Theorem relating the matrix of a composition to the matrices of the individual transformations.
- Theorem concerning the matrix of the identity matrix.
Solve the following problems:[edit]
- Section 2.3, problems 33, 35, 38, 39, 43, 45, 46, and 47.
- Find the matrix representing orthogonal projection onto the $x$-axis. Then find the matrix representing orthogonal projection on the $y$-axis. Finally, find the matrix that represents the composition of these two projections. Does the answer surprise you? Illustrate with a sketch.
- Would the answer to the previous question change if the lines onto which we project were not perpendicular to each other? Illustrate with a sketch.
- Is orthogonal projection on a line an invertible transformation? Why or why not?
