Please answer d, e, f 9:07 PM Tue Dec 117% IT.+: 0The theorems about determinants and areas/volumesof parallelepipeds always consider "full-dimensional"shapes, like 2-dimensional shapes in R², 3-dimensional shapes in R³, and n-dimensional shapesin R". Let's see what happens when we consider alower-dimensional shape in a bigger-dimensionalspace.(a) Let T be the linear transformation from R3 to R3given by orthogonal projection onto the xy-plane.(Then T is represented by the matrix1 0 00 1 0.) What is det A?A =(b) Let N be the 2-dimensional square in R3 definedГоbyand(which lies in the xy-plane). Whatis the area of N?1What is thearea of T(N)?(c) Now let B be the transformation of R3 defined by[10 2 00 03(d) What is det B? 6(e) Let N be the same parallelogram as in part (b). What isthe area of the transformed parallelogram B(N)?6Is the ratio Area(B(N))/Area(2)equal to | det B|? (Write "yes" or "no". Capitalization doesn'tmatter.) yes.So, in summary, let's think more generally:(f) True/false. If N is a 2-dimensional parallelogram inside R3and 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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(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)? Is the ratio Area(B(N))/Area(N) 6 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 SN is a 2-dimensional parallelogram inside R3 and T(ä) = Aa 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)

(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)? Is the ratio Area(B(N))/Area(N) 6 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 SN is a 2-dimensional parallelogram inside R3 and T(ä) = Aa 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)

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 d, e, f

9:07 PM Tue Dec 1
17% I
T.
+: 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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