(a) Let T be the linear transformation from R³ to R³ given by orthogonal projection onto the xy-plane. (Then T is represented by the matrix 1 0 0 A = 0 1 0 1.) What is det A? 0 0 0 (b) Let N be the 2-dimensional square in R3 defined 1 0. by 0 and 1 | (which lies in the xy-plane). What is the area of N? 1 What is the area of T(N)? 0 (c) Now let B be the transformation of R3 defined by 1 0 0 0 2 0 0 0 3 (d) What is det B? 6
(a) Let T be the linear transformation from R³ to R³ given by orthogonal projection onto the xy-plane. (Then T is represented by the matrix 1 0 0 A = 0 1 0 1.) What is det A? 0 0 0 (b) Let N be the 2-dimensional square in R3 defined 1 0. by 0 and 1 | (which lies in the xy-plane). What is the area of N? 1 What is the area of T(N)? 0 (c) Now let B be the transformation of R3 defined by 1 0 0 0 2 0 0 0 3 (d) What is det B? 6
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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Question
Please answer a, b, c
Transcribed Image Text:9:07 PM Tue Dec 1
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+: 0
The theorems about determinants and areas/volumes
of parallelepipeds always consider "full-dimensional"
shapes, like 2-dimensional shapes in R², 3-
dimensional shapes in R³, and n-dimensional shapes
in R". Let's see what happens when we consider a
lower-dimensional shape in a bigger-dimensional
space.
(a) Let T be the linear transformation from R3 to R3
given by orthogonal projection onto the xy-plane.
(Then T is represented by the matrix
1 0 0
0 1 0.) What is det A?
A =
(b) Let N be the 2-dimensional square in R3 defined
Го
by
and
(which lies in the xy-plane). What
is the area of N?
1
What is the
area of T(N)?
(c) Now let B be the transformation of R3 defined by
[1
0 2 0
0 0
3
(d) What is det B? 6
(e) Let N be the same parallelogram as in part (b). What is
the area of the transformed parallelogram B(N)?
6
Is the ratio Area(B(N))/Area(2)
equal to | det B|? (Write "yes" or "no". Capitalization doesn't
matter.) yes
.So, in summary, let's think more generally:
(f) True/false. If N is a 2-dimensional parallelogram inside R3
and T(a) = A is a linear transformation from R3 to R³,
then the area of T(N) is equal to | det A| · (area of N).
(Write "true" or "false". Capitalization doesn't matter)
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