Vector spaces Definition and 286 Threads

Notations: V denotes a vector space A, B, C, D denote subspaces of V respectively ≈ denotes the isomorphic relationship of the left and right operand dim(?) denotes the dimension of "?" Question: Find a vector space V and decompositions: V = A ⊕ B = C ⊕ D with A≈C but B and D are not...

Homework Statement a belongs to R show that the map L: R^n------R^n>0 (R^n>0 denote the n-fold cartesian product of R>0 with itself) (a1) (...) ---------- (an) (e^a1) (...) (e^an) is a isomorphism between the vector space R^n and the vector space R^n>0 Homework Equations...

Homework Statement Let V be the set of all complex-valued functions, f, on the real line such that f(-t)= f(t) with a bar over it, which denotes complex conjugation. Show that V, with the operations (f+g)(t)= f(t)+g(t) (cf)(t)=cf(t) is a vector space over the field of real numbers...

hey can someone suggest me a linear algebra book which dwells into topics of vector spaces(linear) ?

Assignment question: Let V = P (R) and for j >= 1 define T_j(f(x)) = f^j (x) where f^j(x) is the jth derivative of f(x). Prove that the set {T_1, T_2,..., T_n } is a linearly independent subset of L(V) for any positive integer n. I have no idea how...

Hello, very new to vector spaces, it seems like they take some getting used to. Anyway, since spans are sets of all the linear combinations of vectors contained within subspaces, I wonder whether or not vector spaces which contain elements (or vectors) that follow the ten axioms can be...

Homework Statement Knowing this set spans M22: [1 , 0] , [0 , 1] , [0 , 0] ,[0 , 0] [0 , 0] , [0 , 0] , [1 , 0] ,[0 , 1] What is another spanning set for this vector space? Justify your choice by showing that it is a linearly independent set. The Attempt at a Solution [2 , 0] , [0 , 2] , [0...

Homework Statement Which of the following subsets of R3? The set of all vectors of the form a) (a, b, c), where a=c=0 b) (a, b, c), where a=-c c) (a, b, c), where b=2a+1Homework Equations A real vector space is a set of elements V together with two operations + and * satisfying the following...

Homework Statement Let S={A (element) M2(R) : det(A) = 0} (b) Give an explicit example illustrating that S is not closed under matrix addition.Homework Equations The Attempt at a Solution 1) I think that the problem is saying S is a set of 2x2 matrices, whose determinant is zero? 2) I'm...

I had a quick question: Is the following proof of the theorem below correct? Theorem: If C is a convex subset of a Topological vector space X, and the origin 0 in X is contained in C, then the set tC is a subset of C for each 0<=t<=1. Proof: Since C is convex, then t*x + (1-t)*y...

Homework Statement Let V be a vector space over k and S the set of all subspaces of V. Consider the operation of subspace addition in S. Show that there is a zero in S for this operation and that the operation is associative. Consider the operation of intersection in S. Show that this...

Let C([0,1]) be the collection of all complex-valued continuous functions on [0,1]. Define d(f,g)=\int\limits_0^{1}\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}dx for all f,g \in C([0,1]) C([0,1]) is an invariant metric space. Prove that C([0,1]) is a topological vector space

Homework Statement Let F be the field of all real numbers and let V be the set of all sequences (a1,a2,...a_n,...), a_i in F, where equality, addition, and scalar multiplication are defined component-wise. (a) Prove that V is a vector space over F (b) Let W={(a1, a2,...,a_n,...) in V | lim...

1. Suppose V,W are vector spaces over a field F and that T: V ---> W is a linear transformation. Show that for any v belonging to V that T(-v) = -T(v) 2. -T(v) denotes the additive inverse of T(v) 3. I think I'm really overcomplicating it =/ But i have 0v = T( v - v ) = T(v) +...

Homework Statement Let V be a vector space and U a subspace of V . For a given x ∈ V , define T= {x + u | u ∈ U }. Show that T is a subspace of V if and only if x ∈ U . Homework Equations Subspace Test: 1: The 0 vector of V is included in T. 2: T is closed under vector addition 3...

I have never taken linear algebra, but we're doing some catch-up on it in my Quantum Mechanics class. Using teh Griffiths book, problem A.2 if you're curious. Please explain how to solve this, if you help me. If you know of resources on how to think about this stuff, I'd greatly appreciate...

I have never taken linear algebra, but we're doing some catch-up on it in my Quantum Mechanics class. Using teh Griffiths book, problem A.2 if you're curious. Please explain how to solve this, if you help me. If you know of resources on how to think about this stuff, I'd greatly appreciate...

Hello, how to prove this: V^{\bot}\cap W^{\bot}=(V+W)^{\bot} Thanks

sadly not been able to put much effort into this one! was a lecture i missed towards the end of term and didnt get the notes on it, however here is the question. for K>or equal to 1 let Pk denote the the vector space of all real polynomials of degree at most k. For which value of n is Pk...

Greetings, Slowly I am beginning to think that I must be some sort of retard for not getting this fundamental concept. For this post, I will adapt the bracket notation as introduced by P. Halmos' "Finite-dimensional Vector Spaces". \left[ \cdot, \cdot \right] : V \times V^* \to K . A...

I've taken a course in Linear Algebra, so I'm used to working with vector spaces. But now, I'm reading Griffith's Introduction to Elementary Particles, and it talks about groups having closure, an identity, an inverse, and being associative. With the exception of commutativity (unless the...

Hi all! I´m trying to prove following two inequalities but I somehow got stuck: U, W are subspaces of V with dimV = n 1) dimV >= dim(U+W) 2) dim(U+W)>=dimU and dim(U+W)>=dimW Could you give me some hints? thanks in advance!

The only reading on vacuous truth has been from Wikipedia, so I may be misunderstanding something here. Anyway, I was skimming through a Linear Algebra textbook and it said that the empty set is NOT a subspace of every vector space. But I was thinking, shouldn't this be vacuously true? For...

Homework Statement Let V and V' be real normed vector spaces and let f be a linear transformation from V to V'. Prove that f is continuous if V is finite dimensional. The attempt at a solution Let v_1, v_2, \ldots, v_n be a basis for V, let e > 0 and let v in V. I must find a d such that...

How should I represent a vector space V with scalar field F and operation + and x? Is the notation [V, F, +, x] used, or should I use something else?

Homework Statement Let V = {(a1,a2,...an): ai \in C for i = 1,2,...n}; Is V a vector space over the field of real numbers with the operations of coordinatewise addition and multiplication? Homework Equations I know that V is a vector space over C. The Attempt at a Solution I...

Hello, I am currently working out of FDVS - Halmos, and I was wondering if a solutions manual (for the problems at the end of each section) existed? I'd like to be able to check my work. Thanks, Steve P.S Sorry if this is an inappropriate post for this section.

Homework Statement Consider a symmetric (and hence diagonalizable) n x n matrix A. The eigenvectors of A are all linearly independant, and hence they span the eigenspace Rn. Since the matrix A is symmetric, there exists an orthonormal basis consisting of eigenvectors. My questions are...

Would the space C(a,b) (where any element of the space is a continuous complex function) also be a space over the field R of real numbers since the field C has a subfield that is isomorphic to R?EDIT: I am thinking yes because all of the axioms that have to be satisfied in order for a set to be...

Homework Statement Let S be the subspace P3 consisting of all polynomials P(x) such that p(0) = 0, and let T be the subspace of all polynomials q(x) such that q(1) = 0. Find a basis for S, T and S\capT Homework Equations The Attempt at a Solution I know that a basis is formed by...

I'm reading the Wikipedia article, trying to understand the definition of the tensor product V\otimes W of two vector spaces V and W. The first step is to take the cartesian product V\times W. The next step is to define the "free vector space" F(V\times W) as the set of all linear combinations...

Please tell me one of the bases for the infinite dimenional vector space - R (the set of all real numbers) over Q (the set of all rational numbers). The vector addition, field addition and multiplication carry the usual meaning.

I'm having trouble finding a basis for algebraically defined vector spaces where there is more than one condition. For instance, I can easily find a basis for the vector space in R^3 defined by a+2b+3c=0 (where a,b,c are the elements of the vector), but I have no idea what to do when the vector...

1. Homework Statement [/b] For each of the following choices of A and b, determine if b is the column space of A and state whether the system Ax=b is consistent A is a 2 by 2 matrix , or A=(1,2,2,4) , 1 and 2 being on the first row and 2 and 4 on the second row. and b=[4,8] 4 being on...

Homework Statement Let V be a vector space over a field F and let X, Y and Z be a subspaces of V such that X\subseteqY. Show that Y\cap(X+Z) = X + (Y\capZ). (Hint. Show that every element of the LHS is contained on the RHS and vice versa.) Homework Equations The Attempt at a...

[SOLVED] Closed real vector spaces Homework Statement Determine whether the given set V is closed under the operations (+) and (.): V is the set of all ordered pairs of real numbers (x,y) where x>0 and y>0: (x,y)(+)(x',y') = (x+x',y+y') and c(.)(x,y) = (cx,cy), where c is a...

Hey guys, I need to prove a few theorems about vector spaces using the axioms. a) Prove: if -v = v, then v = 0 b) Prove: (-r)v = -(rv) c) Prove: r(-v) = -(rv) d) Prove: v - (-w) = v + w where r is a scalar and v, w are vectors.

[SOLVED] vector spaces + proving of properties Im aware in vector spaces that there are 3 properties associated with it Note v is an element in a vector space, 0 is the additive identity in the vector space and c is a field element 1) 0.v = 0 2) c.0 = 0 3) (-c).v = c.(-.v) = -(c.v)...

Hello all. I am reading again about free vector spaces over a set. In the Theory of Groups by Kurosh part of the construction of a free group is to construct a set of elements inverse to the those of the original set which can effectively "cancel" each other out if juxtaposed in a word made...

I think I have something mixed up so if someone can please point out my error. 1. the set of all linear combinations is called a span. 2. If a family of vectors is linearly independent none of them can be written as a linear combination of finitely many other vectors in the collection. 3. If...

Can someone explain isomorphism to me, with respect to vector spaces. Thanks!

Suppose V and W are vector spaces, and {v1...vn} is basis for V and T. S is an element of L(V, W). Suppose further that T(vi)=S(vi) for all i with 1<= i <=n. Show that S=T. Here's what I think. Because S is an element of L(V,W), S:V-->W means that S has a basis of {v1...vn}, and two...

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...

Homework Statement What is an example of a subset of R^2 which is closed under vector addition and taking additive inverses which is not a subspace of R^2? R, in this question, is the real numbers. Homework Equations I know that, for example, V={(0,0)} is a subset for R^2 that...

I am totally lost on the following questions. What does exhibit mean? 1) Show that the given set H is a subspace of ℜ^3 by finding a matrix A such that N(A) = H (in this case, N(A) represents the null space of A). 2) Exhibit a basis for the vector space H. a b {for all R^3...

I wasn't quite sure where to post this, as it isn't really a homework question. My professor is teaching us General Relativity from a post-grad book, and I don't have a lot of linear algebra under my belt. He lent me the textbook he's teaching from the other day, and I got stuck when I got to...

What exactly is so special about them? What makes a set of vectors that are closed under addition/scalar multiplication and contain 0 so important in math? I've worked through many examples and always wonder... what do these rules mean.

Which vector spaces are isomorphic to R6? a) M 2,3 b) P6 c) C[0,6] d) M 6,1 e) P5 f) {(x1,x2,x3,0,x5,x6,x7)} I know that without showing my work, helper won't answer my question. Since i don't even where to start, all i need is an example. I don't need the complete solution for it. I...

Note: M22 is the set of all m x n matrices with real entries P3 is the set of all polynomials of degree at most n, together with the zero polynomial. 1) Find a basis of M22 consisting of matrices with the property that A^2 = A. I only found 2 of the vectors with a lot of hard work... [1...

Are these two sets: A = {(0,2,2)^T, (1,0,1)^T, (1,2,1)^T} B= {(1,2,0)^T, (2,0,1)^T, (2,2,0)^T} Bases of V3(R) I have found equations that show that they span V3(R) And that both set are linearly independant. So am I right in saying that they are both bases of V3(R). Cheers Ash