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Consider P4 be a set of all the point on a plane through the origin in R“. The general equation of a plane through the origin in R“ is as follow: aw + bx + cy + dz = 0, where a, b, c, and d are fixed constant and at least one is not zero. Show that P4 with the standard addition and scalar multiplication is a vector space.

Consider P4 be a set of all the point on a plane through the origin in R“. The general equation of a plane through the origin in R“ is as follow: aw + bx + cy + dz = 0, where a, b, c, and d are fixed constant and at least one is not zero. Show that P4 with the standard addition and scalar multiplication is a vector space.

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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According this 10 axiom and Create 3 equations that follow the form aw + bx + cy + dz = 0
Vector Space Axioms Definition Let V be a set on which two operations, called vector addition and vector scalar multiplication, have been defined. If u and v are in V , the sum of u and v is denoted by u+v, and if k is a scalar, the scalar multiple of u is denoted by ku. If the following axioms true for all u, v and w in V and for all scalars k and 1, then V is called a vector space and its objects in V are called vectors. 1) u+v is in V 2) u+v=v+u 3) (u+ v)+w =u+(v+w) 4) 0+v= V 5) v+(-v)= 0 6) ku is in V 7) k(u+v)= ku + kv 8) (k +1)u = ku + lu 9) k(lu)=(kl)(u) 10) lv = v QUESTION Consider P4 be a set of all the point on a plane through the origin in R*. The general equation of a plane through the origin in R* is as follow: aw + bx + cy + dz = 0, where a, b, c, and d are fixed constant and at least one is not zero. Show that P4 with the standard addition and scalar multiplication is a vector space.
Transcribed Image Text:Vector Space Axioms Definition Let V be a set on which two operations, called vector addition and vector scalar multiplication, have been defined. If u and v are in V , the sum of u and v is denoted by u+v, and if k is a scalar, the scalar multiple of u is denoted by ku. If the following axioms true for all u, v and w in V and for all scalars k and 1, then V is called a vector space and its objects in V are called vectors. 1) u+v is in V 2) u+v=v+u 3) (u+ v)+w =u+(v+w) 4) 0+v= V 5) v+(-v)= 0 6) ku is in V 7) k(u+v)= ku + kv 8) (k +1)u = ku + lu 9) k(lu)=(kl)(u) 10) lv = v QUESTION Consider P4 be a set of all the point on a plane through the origin in R*. The general equation of a plane through the origin in R* is as follow: aw + bx + cy + dz = 0, where a, b, c, and d are fixed constant and at least one is not zero. Show that P4 with the standard addition and scalar multiplication is a vector space.
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