3. In this exercise you will explore projection mappings. (a) Projections onto lines in R2: Let T: R² R² be the linear transformation which projects vectors onto the x-axis. i. Sketch the unit square in R2 and the image of the unit square under T on a separate set of axis. ii. Using only a geometric argument (i.e. no matrices or computations) find the eigenvalues and the basic eigenvectors of AT. iii. Is AT diagonalizable? Justify your answer. iv. Find the standard matrix for AT. (b) Projections onto planes in R³: Let T: R³ which projects vectors onto the xy-plane. R³ be the linear transformation i. Sketch the unit cube in R3 and the image of the unit cube under T on a seperate set of axis. ii. Using only a geometric argument (i.e. no matrices or computations) find the eigenvalues and the basic eigenvectors of AT. iii. Is Ar diagonalizable? Justify your answer. iv. Find the standard matrix for AT. (c) Projections in R4: In part (a) we saw an example of projecting from R2 to R, and in (b) we saw an example of projecting from R³ to R2. Now consider the case of R4. The 4-dimensional unit cube in R4 has vertices (±1, +1, +1, +1). Following the work above, the natural way to project the unit cube into R³ is by using the 100 0 1 0 0 00 10 matrix A = . The output of the 4-dimensional unit cube under TA, is 0 00 0 the 3-dimensional object pictured below. It's called the tessaract. This fascinating 3-dimensional object is the 'shadow of the unit cube from R4, something which is beyond our scope of imagination! i. List the eigenvalues of A and their basic eigenvectors. (There are no tricks here, this is an easy exercise!) ii. Is A diagonalizable? Justify your answer. (There are no tricks here, this is an easy exercise!)
3. In this exercise you will explore projection mappings. (a) Projections onto lines in R2: Let T: R² R² be the linear transformation which projects vectors onto the x-axis. i. Sketch the unit square in R2 and the image of the unit square under T on a separate set of axis. ii. Using only a geometric argument (i.e. no matrices or computations) find the eigenvalues and the basic eigenvectors of AT. iii. Is AT diagonalizable? Justify your answer. iv. Find the standard matrix for AT. (b) Projections onto planes in R³: Let T: R³ which projects vectors onto the xy-plane. R³ be the linear transformation i. Sketch the unit cube in R3 and the image of the unit cube under T on a seperate set of axis. ii. Using only a geometric argument (i.e. no matrices or computations) find the eigenvalues and the basic eigenvectors of AT. iii. Is Ar diagonalizable? Justify your answer. iv. Find the standard matrix for AT. (c) Projections in R4: In part (a) we saw an example of projecting from R2 to R, and in (b) we saw an example of projecting from R³ to R2. Now consider the case of R4. The 4-dimensional unit cube in R4 has vertices (±1, +1, +1, +1). Following the work above, the natural way to project the unit cube into R³ is by using the 100 0 1 0 0 00 10 matrix A = . The output of the 4-dimensional unit cube under TA, is 0 00 0 the 3-dimensional object pictured below. It's called the tessaract. This fascinating 3-dimensional object is the 'shadow of the unit cube from R4, something which is beyond our scope of imagination! i. List the eigenvalues of A and their basic eigenvectors. (There are no tricks here, this is an easy exercise!) ii. Is A diagonalizable? Justify your answer. (There are no tricks here, this is an easy exercise!)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:3. In this exercise you will explore projection mappings.
(a) Projections onto lines in R² : Let T : R² → R² be the linear transformation which
projects vectors onto the x-axis.
i. Sketch the unit square in R² and the image of the unit square under T on a
separate set of axis.
ii. Using only a geometric argument (i.e. no matrices or computations) find the
eigenvalues and the basic eigenvectors of AT.
iii. Is AȚ diagonalizable? Justify your answer.
iv. Find the standard matrix for AT.
(b) Projections onto planes in R³ : Let T : R³ → R³ be the linear transformation
which projects vectors onto the xy-plane.
i. Sketch the unit cube in R³ and the image of the unit cube under T on a seperate
set of axis.
ii. Using only a geometric argument (i.e. no matrices or computations) find the
eigenvalues and the basic eigenvectors of AT.
iii. Is AȚ diagonalizable? Justify your answer.
iv. Find the standard matrix for AT.
(c) Projections in R4: In part (a) we saw an example of projecting from R² to R,
and in (b) we saw an example of projecting from R³ to R². Now consider the case
of R4. The 4-dimensional unit cube in R4 has vertices (±1, ±1, ±1, ±1). Following
the work above, the natural way to project the unit cube into R³ is by using the
000
0100
001
0 00 0
the 3-dimensional object pictured below. It's called the tessaract. This fascinating
3-dimensional object is the 'shadow' of the unit cube from R4, something which is
beyond our scope of imagination!
matrix A
The output of the 4-dimensional unit cube under TA, is
i. List the eigenvalues of A and their basic eigenvectors. (There are no tricks here,
this is an easy exercise!)
ii. Is A diagonalizable? Justify your answer. (There are no tricks here, this is an
easy exercise!)
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