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 Rª 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.

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 Rª 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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

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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+l)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 Rª. The general equation of a
plane through the origin in Rª 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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