Theorem (7.5) Let S and T be finitely generated subspaces of a vector space V. Then S n T and S + T are finitely generated subspaces, and we havedim(S + T) + dim (S n T) = dim S + dim T 5. / Letaz = <2, 1, 0, – 1)az = (1, – 3, 2, 0>az = (4, 8, – 4, – 3> a4 = <1, 10, – 6, – 2> as = <3, – 1, 2, 4>as = <-2, 0, 6, 1>%3D%3Dand letS = S(a1, az, a3 , as),T = S(a4, as , as).%3DFind dim S, dim T, dim (S + T), and, using Theorem (7.5), find dimYsn T).

According to the given information it is required to find dim S, dim T, dim(S+T), and using theorem,…

ai = (2, 1, 0, –1> <4, 8, – 4, – 3> as = <-2, 0, 6, 1> az = (1, – 3, 2, 0> az = a, = <1, 10, –6, – 2> ae = <3, – 1, 2, 4> (3, - 1, 2, 4> and let S = S(a1, a2, a3, as), T = S(a4, as, as) . %3D Find dim S, dim T, dim (S + T), and, using Theorem (7.5), find dim Ys n T).

ai = (2, 1, 0, –1> <4, 8, – 4, – 3> as = <-2, 0, 6, 1> az = (1, – 3, 2, 0> az = a, = <1, 10, –6, – 2> ae = <3, – 1, 2, 4> (3, - 1, 2, 4> and let S = S(a1, a2, a3, as), T = S(a4, as, as) . %3D Find dim S, dim T, dim (S + T), and, using Theorem (7.5), find dim Ys n T).

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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Theorem (7.5) Let S and T be finitely generated subspaces of a vector space V. Then S n T and S + T are finitely generated subspaces, and we have

dim(S + T) + dim (S n T) = dim S + dim T

5. / Let
az = <2, 1, 0, – 1)
az = (1, – 3, 2, 0>
az = (4, 8, – 4, – 3> a4 = <1, 10, – 6, – 2> as = <3, – 1, 2, 4>
as = <-2, 0, 6, 1>
%3D
%3D
and let
S = S(a1, az, a3 , as),
T = S(a4, as , as).
%3D
Find dim S, dim T, dim (S + T), and, using Theorem (7.5), find dim
Ysn T).

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