Vector space Definition and 540 Threads

Here is my attempt to answer this guys, i'd really appreciate any corrections. a vector space has the 0 vector the vector space is closed under vector addition and scalar multiplication (AKA for every vector u, v in the subspace, there exists a vector u + v in the subspace) Here we go...

I came across this problem today and haven't been able to figure it out... Give an example of a vector space V which isomorphic to a proper subspace W, i.e. V != W. It seems to me that V can't have a finite basis, but can't think of any examples regardless...any thoughts?

I am trying to shorten and generalize the the definition of a vector space to redefine it in such a way that only four axioms are required. The axioms must hold for all vectors u, v and w are in V and all scalars c and d. I believe the four would be: 1. u + v is in V, 2. u + 0 = u 3. u...

Hello all. While looking at vector spaces leading up to multilinear mappings i am having trouble right at the start with the idea of the set of all real valued functions on an arbitrary set which vanish at all but a finite number of points. The author ( Wasserman, Tensors and Manifolds ) does...

Homework Statement Determine whether the following 2 x 2 matrices form a basis of the vector space M2x2(R) of all (2x2)-matrices over R: A1= 1 0 0 0 A2= 2 2 0 0 A3= 3 2 1 0 A4= 4 3 2 1 Homework Equations The Attempt at a Solution So for them to be...

Hello all I have read about quotient spaces of a vector space in several books and have an understanding of what they are. Looking up Quotient Vector Space in Wiki it says :- The quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. I...

Homework Statement V is vector space of all antisimetric 3x3 matrices. Find the coordinates of the matrix A= \left| \begin{array}{ccc} \ 0 & 1 & -2 \\ -1 & 0 & -3 \\ 2 & 3 & 0 \end{array} \right|\] relative to the base E_1= \left|...

Hello Well i hv just started vector spaces and well am finding difficulty in proving this hoffman and kunze problem could some 1 help me :( Question : W1 and W2 be sub-spaces of vector space V such that set-theoritic union of W1 and W2 is also a Subspace . Proove that one of the subspace...

Homework Statement Let R denote the set of real numbers. Define scalar multiplication by \alpha x = \alpha x which is simply regular scalar multiplication, and vector addition is defined as x \oplus y = max(x,y). Is R a vector space with these operations? Homework Equations Some given...

Homework Statement Im doing a problem where I am trying to show that an abelian group with a scalar multiplication is a vector field. I am trying to show associativity right now and just have a question: im trying to show that exp(b.c.lnx) = b.exp(c.lnx) But I am not very sure of my logs...

Homework Statement Show that F[x]/( g(x) ) is a n-dimensional vector space. where g is in F[x], and g has degree n. Its clear that F[x]/( g(x) ) is a vector space and that B= (1,x^{2},...,x^{n-1}) spans F[x]/( g(x) ), but I am having trouble showing that B is linearly independent...

Homework Statement Show that F[x]/( g(x) ) is a n-dimensional vector space. where g is in F[x], and g has degree n. Its clear that F[x]/( g(x) ) is a vector space and that B= (1,x^{2},...,x^{n-1}) spans F[x]/( g(x) ), but I am having trouble showing that B is linearly independent

Question 1 Let u, v1,v2 ... vn be vectors in R^{n}. Show that if u is orthogonal to v1,v2 ...vn then u is orthogonal to every vector in span{v1,v2...vn} My attempt if u is orthogonal to v1,v2 ...vn then (u.v1)+(u.v2)+...+(u.vn)=0 Let w be a vector in span{v1,v2...vn} therefore...

I'm just wondering what are the differences between vector spaces and fields. From what I understand by the definitions, both of these are collections of objects where additions and scalar multiplications can be performed. I can't seem to see the difference between vector spaces and fields.

Consider the Minkowski space of 4 dimensions with signature (- + + +). How does the vector space algebra work here? More specifically given 3 space like orthonormal vectors how do we define fourth vector orthogonal to these vectors? I am looking for an appropriate vector product like it is in...

Hello all. Back to basics again. When defining a set of geometric vectors for a vector space of n dimensions how can we define such a set without a certain amount of structure already defined upon the n dimensional space. We presumably need some concept of direction to determine linear...

Homework Statement Do functions that vanish at the end points x=0 and x=L form a vector space ? How about periodic functions obeying f(L)=f(0)? How about functions that obey f(0)=4 ? If the functions do not qualify , list the things that go wrong . Homework Equations All axioms...

Maximal subspace Problem: Prove that every vector space V has maximal subspace, i.e. a proper subspace that is not properly contained in a proper subspace of V. I let A be the collection of all proper subspaces of V, but I can't prove that every totally ordered subcollection of A has an...

concept of a "basis" for a vector space I do not understand the concept of a "basis" for a vector space. Here's an example from my practice final exam: Suppose U and V are subspaces of the real vector space W and {u1} is a basis for U and {v1} is a basis for V. If U intersection V = {0}...

Homework Statement Let {W_1,W_2,W_3,...} be a collection of proper subspaces of V (i.e. W_i not=V) such that W_i is a subset of W_(i+1) for all i. Prove that U(W_i) (i from 1 to infinity) is a proper subspace of V The Attempt at a Solution I've already proven that U(W_i) is a subspace of...

Hello all. I came across this problem in Halmos, Finite-Dimensional Vector Spaces, page 16. Is the set R of all real numbers a finite-dimensional vector space over the field Q of all rational numbers. There is a reference to a previous example which says that with the usual rules of...

[SOLVED] a simple vector space problem Homework Statement Consider the set of all entities of the form (a,b,c) where the entries are real numbers . Addition and scalar multiplication are defined as follows : (a,b,c) + (d,e,f) = (a+d,b+e,c+f) z*(a,b,c) = (za,zb,zc) Show that vectors...

I am reading "The linear algebra a beginning graduate student ought to know" by Golan, and I encountered a puzzling statement: Let V be a vector space (not necessarily finitely generated) over a field F. Prove that there exists a bijective function between any two bases of V. Hint: Use...

I'm trying to solve a problem ice109 recommended. I'm trying to show how y=2x+1 is not a vector space. Here I go. Let u=(x,2x+1) v=(x',2x'+1) w=(x",2x"+1) 1. If u and v are objects in V, then u + v is in V. u+v=(x+x',2x+2x'+2) fails because 2 is not in V? 2. u + v = v + u passes...

[b]1. Consider the vector space of polynomials 1+x^3 , 1-x+x^2, 2x, 1+x^2 Are they linearly dependent or independent? dimension of vecotr space spanned by these vectors? [b]3. I have tried to solve this by letting a1 = 1+x^3 a2 = 1-x+x^2 a3 = 2x a4 = 1+x^2 Then I let (alpha)a1 +...

1. Homework Statement A set of objects is given, together with operations of addition and scalar multiplication. Determine which sets are vector spaces under the given operations. For those that are not vector spaces, list all axioms that fail to hold. (x,y,z) + (x',y',z') =...

W={(x1,x2,x3):x^{2}_{1}+x^{2}_{2}+x^{2}_{3}=0} , V=R^3 Is W a subspace of the vector space? from what i understand for subspace to be a subspace it has to have two conditions: 1.must be closed under addition 2.must be closed under multiplication so... I pick a vector s=(s1,s2,s3) and a second...

Hello everyone. I came across the term free vector space in a book on mathematical physics by Geroch but cannot find them in any other of my books. Can someone give me an explanation of how a free vector space differs from a standard vector space. Geroch says that any set can be made into a...

15. Determine wheter the set is a vector space. The set of all fifth-degree polynomials with the standard operations. AXIOMS 1.u+v is in V 2.u+v=v+u 3.u+(v+w)=(u+v)+w 4.u+0=u 5.u+(-u)=0 6. cu is in V 7.c(u+v)=cu+cv 8.(c+d)u=cu+cd 9.c(du)=(cd)u 10.1(u)=u the back of my book says that axioms...

15. Determine wheter the set is a vector space. The set of all fifth-degree polynomials with the standard operations. AXIOMS 1.u+v is in V 2.u+v=v+u 3.u+(v+w)=(u+v)+w 4.u+0=u 5.u+(-u)=0 6. cu is in V 7.c(u+v)=cu+cv 8.(c+d)u=cu+cd 9.c(du)=(cd)u 10.1(u)=u the axioms that fail are...

Linear Algebra: Vector Space proof... I'm really having trouble comprehending this problem. This is not exactly a "homework problem" but I need a good, formal definition of this to help with some other problems. Let (Vectors) V1, V2,...,Vk be vectors in vector space V. Then the set W of all...

Homework Statement Show that the space of all shift maps is indeed a vector space over R and that there is a linear bijection between it and R2 Homework Equations 10 Axioms of vector spaces Definition of bijection (1-1, onto) For 1-1: f(a) = f(b) -> a = b. The Attempt at a...

If G\subset \textrm{End}(V), and W\subset V is a subspace of a vector space V, and somebody says "G leaves W stable", does it mean GW=W or GW\subset W or something else?

Hello all, I've just learned a bit about the complexification of a real vector space V to include scalar multiplication by complex numbers. A bit of confusion has ensued, which I am hoping someone can help me with conceptually: 1) how does one generate a basis for the new space Vc? It seems...

Homework Statement I have been going through some past exam papers and have come across this vector space question that I cannot find relevant examples for. Consider the vector space V of n-th order polynomials p(x) = a0 + a1x + a2x^2 +· · ·+anx^n, where a0,a1,a2, ...,an are real numbers, and...

Homework Statement I'm trying to understand why \ell_2^\infty as a vector space over \mathbb{C}, has uncountable dimension. Homework Equations The Attempt at a Solution Firstly, I'm not really clear on the meaning of basis in infinite dimensions. Is it still true that any element...

Homework Statement Is there any difference between the vector space spanned by the set cos(t),sin(t) and the vector space spanned by the set cos(t)+sin(t),cos(t)-sin(t)? Homework Equations The Attempt at a Solution Not really a homework question but it will help me answer a...

Don't understand this reasoning with respect to linear operators. Let S and T be linear operators on the finite dimensional vector space V. Then assuming the composition ST is invertible, we get \text{null} \; S \subset \text{null} \; ST Why is that? I thought hard about it but I simply...

So I have an assignment due in a few hours and I am pretty happy with it, aside from the fact that I am completely lost on the following section: - The polynomials of degree 3, denoted P3, form a vector space. 1. Show that when added, two general polynomials of degree 3 will always produce...

I am in a problem seminar class and I have not taken Linear Algebra in over 4 years so I am having a lot of problems with this. Please help...:eek: Homework Statement Let P be the set of all polynomials with real coefficients and of degree less than 3. Thus, P = {f:f(x)= a(sub0)...

Hi everyone, Can anyone explain the following to me? Given a basis beta for an n-dimensional vector space V over the field F, "the standard representation of V with respect to beta is the function phi_beta(x)=[x]_beta for each x in V." This is from my textbook. It then proceeds to give...

Homework Statement Show that the positive quadrant Q = ( (x,y) | x,y > 0 ) \in \mathbb{R}^2 is a vector space. Homework Equations Addition is redefined by (x_1,y_1) + (x_2,y_2) = (x_1 x_2, y_1 y_2) and scalar multiplication by c(x,y) = (x^c , y^c) The Attempt at a Solution There...

let V be a vector space and K a nonempty subset of V prove/disprove : K is linear independent set iff for every T such that T is a proper subset of K, span(T) is a proper subset of spanK. im having difficulty finding a counter example, so i think this statement is correct, but how to prove...

Homework Statement Let V=Mn(F) be the space of all nxn matrices over F; define TA=(1/2)(A+transpose(A)) for A in V. Verify that T is not only a linear operator on V, but is also a projection. Homework Equations A is a projection when A squared=A. The Attempt at a Solution I don't...

What is the different between a vector space and a field? Seems to me that they both are the same thing

Please help me proove the following: Let V be a vector space over all n-by-n square matrices. Let W be a non-trivial subspace of V satisfying the following condition: if A is an element of W and B is an element of V then AB, BA are both elements of W. Proove that W = V. And here is what...

Homework Statement One of the fundamental axioms that must hold true for a set of elements to be considered a vector space is as follows: 1*x = x I was given a particular space: The set of all polynomials of degree greater than or equal to three, and zero, and asked to evaluate whether or...

What vector space are cross products done in?

Vector space help please.. Hi, Just started a linear algebra course recently but I am confused with the notation used :confused: http://i9.tinypic.com/2w4za50.jpg I am unsure how to proceed with this question. Can someone help? The part highlighted, what does it mean? 2x2 matrix of...

I am supposed to determine whether or not the following two sets constitute a vector space. 1) The set of all polynomials degree two. 2) The set of all diagonal 2 x 2 matrices. For the first one, it will not be a vector space because it does not satisfy the closure property. Also the...