Consider the vectors u= 1 V= W= and let H be the subset of R4 consisting of vectors of the form x = u + rv + sw+tz with r, s, t € R. (a) If x= [1, 22, 23, 24] is in H, show that 8r+ 6s + 5t=2₁-6 2r + 2s+ t = 2₂ - 1 r+2s+ t = x3 - 1 r+ s + t = x4 - 1 (b) Put the system in part (a) in row-echelon form, where the variables in the system are r, s, t (rather than 21, 22, 23, 24). (c) Your row-echelon form will look like 0 * 00 000 Z= * #### where each is a non-zero real number, each is a real number, and each # is an expression in 21, 22, 23, 24. Use your row-echelon form to find a condition on 21, 22, 23, 24 for the system to have a solution [r, s, t], and thus find an equation of the form a121 + a222 + a3x3 + a4x4 = b defining H.
Consider the vectors u= 1 V= W= and let H be the subset of R4 consisting of vectors of the form x = u + rv + sw+tz with r, s, t € R. (a) If x= [1, 22, 23, 24] is in H, show that 8r+ 6s + 5t=2₁-6 2r + 2s+ t = 2₂ - 1 r+2s+ t = x3 - 1 r+ s + t = x4 - 1 (b) Put the system in part (a) in row-echelon form, where the variables in the system are r, s, t (rather than 21, 22, 23, 24). (c) Your row-echelon form will look like 0 * 00 000 Z= * #### where each is a non-zero real number, each is a real number, and each # is an expression in 21, 22, 23, 24. Use your row-echelon form to find a condition on 21, 22, 23, 24 for the system to have a solution [r, s, t], and thus find an equation of the form a121 + a222 + a3x3 + a4x4 = b defining H.
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
![Consider the vectors
u=
V =
W =
8r+ 6s+ 5t = 21 - 6
2r + 2s+ t =
x₂ - 1
and let H be the subset of R4 consisting of vectors of the form
x=u+rv+sw+tz with r, s, t e R.
(a) If x= [1, 22, 23, 24] is in H, show that
r+2s+ t =
r+ s +
23 - 1
t = xX4 - 1
*
F
(b) Put the system in part (a) in row-echelon form, where the variables in
the system are r, s, t (rather than ₁, X2, X3, T4).
(c) Your row-echelon form will look like
*
0
00
000 #
###:
Z =
where each * is a non-zero real number, each is a real number, and each
# is an expression in 21, 22, 23, 24. Use your row-echelon form to find
a condition on 21, 22, 23, 24 for the system to have a solution [r, s, t],
and thus find an equation of the form a121 + a₂x2 + a3x3 + a4x4 = b
defining H.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87351664-192b-44e9-8681-fefec31a9dea%2Fe27b91ce-2c4b-4f33-9a52-463b5172996c%2Fiu3twcy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the vectors
u=
V =
W =
8r+ 6s+ 5t = 21 - 6
2r + 2s+ t =
x₂ - 1
and let H be the subset of R4 consisting of vectors of the form
x=u+rv+sw+tz with r, s, t e R.
(a) If x= [1, 22, 23, 24] is in H, show that
r+2s+ t =
r+ s +
23 - 1
t = xX4 - 1
*
F
(b) Put the system in part (a) in row-echelon form, where the variables in
the system are r, s, t (rather than ₁, X2, X3, T4).
(c) Your row-echelon form will look like
*
0
00
000 #
###:
Z =
where each * is a non-zero real number, each is a real number, and each
# is an expression in 21, 22, 23, 24. Use your row-echelon form to find
a condition on 21, 22, 23, 24 for the system to have a solution [r, s, t],
and thus find an equation of the form a121 + a₂x2 + a3x3 + a4x4 = b
defining H.
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