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

See similar textbooks

Related questions

Q: Use proof by contradiction to prove the statement.In a vector space, the zero vector is unique.

Q: Let (:/ be an inner product in the vector space V. Given an isomorphism1' : U + [u, v] = (Tu, Tv),…

A: Let , be an inner product in the vector space V and given an isomorphism T : U→V.…

Q: - Let V be the set of all ordered pairs of real numbers under the following operations of addition…

Q: 1. Let, p is the set of all vectors of the form: m,n, te R. 2n (a) Is p closed under vector…

Q: Let V be the set of all positive real numbers. Determine whether V is a vector space with the…

A: V=x∈R+ which is set of positive real numbers addition operation x+y=xy scalar multiplication cx=xc

Q: Linear Algebra

A: A space comprised of vectors, collectively with the associative and commutative law of addition of…

Q: Question (3): Determine whether V is a vector space. If it is, verify each vector space axioms; if…

A: Let V={(x,y) | x, y ∈R and x≥0} To check if this is a vector space or not. This can be done as…

Q: 2. Determine whether the set of all triples of real numbers with addition defined by (x, y, z) + (u,…

Q: Determine if S=R (real numbers) where -1 is not included, is a vector space with the operations:…

A: We determine whether S is a vector space with the given operations.

Q: . Let V be a vector space, and {v1, ··· , vn} be a set of vectors that spans V . Show that the set…

Q: Zero is the only vector in an inner product space V that is orthogonal to itself. O True False

A: The statement is true. We give the justification in the next step.

Q: Algebra Suppose {v1, v2, V3} is a linearly independent set of vectors in R". (a) Give a lower bound…

Q: Let V be the set of all positive real numbers. Determine whether V is a vector space with the…

Q: If vi, v2, ..., Vn are independent vectors in R", then they must form a basis of R". True False…

Q: Let V be set of all ordered pairs of real numbers and consider the following addition and scalar…

A: Given: u=(0,4) v=(1,-3) k=2

Q: 1. The given set with the given operations is not a vector space. Check all the properties and list…

Q: Let V₁, V2, V3 be the vectors in R³ defined by 16 11 3 12 34 (a) Is {V₁, V₂, V3} linearly…

Q: Determine a basis for the vector space spanned by the vectors 3. V1 2 ,V2 .Vs 17 12 O A. {V1, V3,…

Q: Given the set of vectors determine whether the set of vectors is a basis for R

Q: be rational numbers. Show that R is a vector space over the rational numbers Q, using standard…

A: We proved all the properties of a vector space.

Q: Show that the set of complex number C={x+iy|x and y are real, i2=-1} is a vector space. Verify each…

Q: Determine if S=R (real numbers) where -1 is not included, is a vector space with the operations:…

A: S does not form vector space.

Q: linear algebra Be V the set of all ordered pairs of real numbers and consider the addition and…

Q: of complex numbers C a vec

Q: = Let C be a set of all complex numbers (i.e., numbers of the form z = a +ib, where a and b are real…

A: Set of complex numbers C defined asC={a+ib:a,b∈R}Set of scalar is field of complex numbers.From the…

Q: The set of m x n matrices is not a vector space under matrix addition and scalar multiplication.…

Q: Determine whether the given sets with given operations are vector spaces? For those that are not…

Q: 7. Show that the set Z of integers is NOT a vector space.

Q: Prove that if {v_1,...,v_n} spans V, then so does {v_1 − v_2, v_2 − v_3,...,v_(n−1) − v_n, v_n}.

Q: Which of the following are axioms of a vector space V? (Select all that are correct.) 1x = x If k,…

Q: Every finitely generated vector space has a finite basis.

A: Every finitely generated vector space has a finite basis.

Q: nd a basis for the set of all vectors of the form below, where a, b, c are real numbers. [ a-2b+5c,…

Q: Let V be the set of all positive real numbers Determine whether V is a vector space with the given…

Concept explainers

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.

Question

100%

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.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

Step 8

Step 9

Step 10

Trending now

This is a popular solution!

Step by step

Solved in 10 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Vector Space$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

X Y x + y Let W be the set of all vectors with x and y real. Find a basis of W 1
Let V be the set of all positive real numbers Determine whether V is a vector space with the given operations. X +Ỷ = XỶ cX = X° if it is verify axioms 4, 8,9.
Let W be the set of all vectors of the form shown on the right, where a and b represent arbitrary real numbers. Find a set S of vectors that spans W, or give an example or an explanation showing why W is not a vector space. -a +3 a-4b 3b + a Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The set W is a vector space and a spanning set is S= (Use a comma to separate vectors as needed.) B. The set W is not a vector space because not all vectors u in W have the property 1u= u. O C. The set W is not a vector space because the zero vector is not in W. O D. The set W is not a vector space because not all vectors u, v, and w in W have the property that u + v=v+u and (u+v)+w=u+(v + W).
Linear algebra
(a) Let V be R², and the set of all ordered pairs (x, y) of real numbers. Define an addition by (a, b) + (c,d) = (a + c, 1) for all (a, b) and (c,d) in V. Define a scalar multiplication by k · (a, b) = (ka, b) for all k E R and (a, b) in V. . Verify the following axioms: (i) k(u + v) = ku + kv (ii) u + (-u) = 0
Let v = {(; Define addition and scalar multiplication as follows. 1 b +e kb kc verify the following axioms: a) (u + v) + w = u + (v + w) b) u + (-u) = –u + u = 0 c) c. (u + v) = (c.u) + (c.v) Note: u, v, w, are vectors and c and d are scalars
The set of complex numbers is a vector space under usual addition and multiplication. Find the negative vector the vector u = a + bi where and a, b are real numbers. -a + bi O a + bi -a – bi Оа — bi
Let W be the set of all vectors of the form shown on the right, where a and b represent arbitrary real numbers. Find a set S of vectors that spans W, or give an example or an explanation showing why W is not a vector space. -a +5 a-3b 5b + a Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The set W is a vector space and a spanning set is S= (Use a comma to separate vectors as needed.) O B. The set W is not a vector space because not all vectors u in W have the property 1u= u. O C. The set W is not a vector space because not all vectors u, v, and w in W have the property that u + v=v+u and (u + v) +w=u+(v +w). O D. The set W is not a vector space because the zero vector is not in W.