4. Let (*) be the following homogeneous system of (n-1) linear equations in n unknows F: All X₁ + A₁₂ X₂ + + An Xn = 0. A₂1 X₁ + A₂₂X₂ + ... + Cl₂n Xn = 0. a22X2 7 An-1,1X₁+ an-1,2 X₂ + + Olm-1.n Xn = 0. over it! (*) Is Suppose that no equation in this system linear ambination of the other equations. For ea =1,2,.., n, let Ai & Max) (F) be the matrix obtained by deleting the ith column of t coefficient matrix of (*), and let (i = (-1) det (Ai).
4. Let (*) be the following homogeneous system of (n-1) linear equations in n unknows F: All X₁ + A₁₂ X₂ + + An Xn = 0. A₂1 X₁ + A₂₂X₂ + ... + Cl₂n Xn = 0. a22X2 7 An-1,1X₁+ an-1,2 X₂ + + Olm-1.n Xn = 0. over it! (*) Is Suppose that no equation in this system linear ambination of the other equations. For ea =1,2,.., n, let Ai & Max) (F) be the matrix obtained by deleting the ith column of t coefficient matrix of (*), and let (i = (-1) det (Ai).
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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Transcribed Image Text:4. Let (*) be the following homogeneous system
F:
of (n-1) linear equations in n unknows
All X₁ + A₁₂ X₂ t
+ An Xn = 0.
a22X2
A21X₁ + a₂₂X₂ + ... + Cl₂n Xn = 0.
An-1,1X₁+ an-1, 2X₂ + + On-in Xn = 0.
,2
Is
Suppose that no equation in this system
linear combination of the other equations. For each
=1,2,.nlet Ai & Ma-x) (F) be the
matrix obtained by deleting the ith column of the
coefficient matrix of (*), and let
det (Ai).
Ci
over
it!
=
(*)
(-1)
Prove the following statements
(a) Not all of C₁,...,
Cn are 0.
(5) The solution space of the system (*) is olle-dimen
(C) A
vector (X₁. --, Xn) € F" is a
-sional,
solution to the
system (*) iff (X₁...., Xn) = t (C₁,
, (n)
a
***/
for some
te F.
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