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(j) Display a non-zero vector in R³ that is neither a solution of the inhomo- geneous system nor the homogeneous system. (k) Show that any solution of the inhomogeneous system€ S can be ex- pressed as (xo, Yo, 20) + (0,0,3), where (ro, yo, zo) € So is a solution of the homogeneous system. (1) How would you describe the geometric relationship between S and So?

(j) Display a non-zero vector in R³ that is neither a solution of the inhomo- geneous system nor the homogeneous system. (k) Show that any solution of the inhomogeneous system€ S can be ex- pressed as (xo, Yo, 20) + (0,0,3), where (ro, yo, zo) € So is a solution of the homogeneous system. (1) How would you describe the geometric relationship between S and So?

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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Could I get answer for part j k and l?

(i) Show that we can add and take scalar multiples of the vectors in So and we stay in So. (j) Display a non-zero vector in R³ that is neither a solution of the inhomo- geneous system nor the homogeneous system. (k) Show that any solution of the inhomogeneous system ES can be ex- pressed as (xo, Yo, 20) + (0, 0,3), where (ro, Yo, zo) So is a solution of the homogeneous system. (1) How would you describe the geometric relationship between S and So?
Transcribed Image Text:(i) Show that we can add and take scalar multiples of the vectors in So and we stay in So. (j) Display a non-zero vector in R³ that is neither a solution of the inhomo- geneous system nor the homogeneous system. (k) Show that any solution of the inhomogeneous system ES can be ex- pressed as (xo, Yo, 20) + (0, 0,3), where (ro, Yo, zo) So is a solution of the homogeneous system. (1) How would you describe the geometric relationship between S and So?
For vectors (x, y, z)¹ € 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 Mã = 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 = {(s, t, 3 s+t): s₁ t ≤ R} CR³. s, (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 z, yE S show that ar + by € S whenever a + b = 1. (f) Give at least one reason why S is not a subspace of R³. (g) Let S C R³ be the set of solutions for the homogeneous system. Give a description of So as a set (similar to that given for S in (c) above). (h) Can So be described geometrically as a plane or a line? If so, which?
Transcribed Image Text:For vectors (x, y, z)¹ € 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 Mã = 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 = {(s, t, 3 s+t): s₁ t ≤ R} CR³. s, (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 z, yE S show that ar + by € S whenever a + b = 1. (f) Give at least one reason why S is not a subspace of R³. (g) Let S C R³ be the set of solutions for the homogeneous system. Give a description of So as a set (similar to that given for S in (c) above). (h) Can So be described geometrically as a plane or a line? If so, which?
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