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(c) Show that the set of solutions of the inhomogeneous system is given by S = {(s, t, 3-s+t): s₁ t € R} CR³. (d) Can S be described geometrically as a plane or a line? If so, which? (e) Explicitly confirm that S is closed under 'affine combinations'. That is, for all r, yE S show that ar + by ES whenever a +b = 1. (f) Give at least one reason why S is not a subspace of R³.

(c) Show that the set of solutions of the inhomogeneous system is given by S = {(s, t, 3-s+t): s₁ t € R} CR³. (d) Can S be described geometrically as a plane or a line? If so, which? (e) Explicitly confirm that S is closed under 'affine combinations'. That is, for all r, yE S show that ar + by ES whenever a +b = 1. (f) Give at least one reason why S is not a subspace of R³.

Advanced Engineering Mathematics
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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I need solution to d,e,f parts thanks. or at least d and e

For vectors (x, y, z) R³ consider the inhomogeneous linear system: x-y+z=3, and its 'associated' homogeneous linear system: x=y+z=0. (a) Express both systems in matrix form M = č. (b) Show that (0, 0, 3) is a solution of the inhomogeneous system. (c) Show that the set of solutions of the inhomogeneous system is given by = {(s, t,3-s+t)¹ : s,t = R} CR³. (d) Can S be described geometrically as a plane or a line? If so, which? (e) Explicitly confirm that S is closed under 'affine combinations' That is, for all , y = S show that aữ + by € S whenever a + b = 1. (f) Give at least one reason why S is not a subspace of R³.
Transcribed Image Text:For vectors (x, y, z) R³ consider the inhomogeneous linear system: x-y+z=3, and its 'associated' homogeneous linear system: x=y+z=0. (a) Express both systems in matrix form M = č. (b) Show that (0, 0, 3) is a solution of the inhomogeneous system. (c) Show that the set of solutions of the inhomogeneous system is given by = {(s, t,3-s+t)¹ : s,t = R} CR³. (d) Can S be described geometrically as a plane or a line? If so, which? (e) Explicitly confirm that S is closed under 'affine combinations' That is, for all , y = S show that aữ + by € S whenever a + b = 1. (f) Give at least one reason why S is not a subspace of R³.
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