Vector spaces Definition and 286 Threads

I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ... I am focused on Section 2.1 Introduction to Linear Transformations ... ... I need help with understanding Theorem 2.7 ... Theorem 2.7, its proof and some remarks read as follows:I am having considerable trouble...

I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ... I am focused on Section 1.6 Bases and Finite-Dimensional Vector Spaces ... I need help with the proof of Theorem 1.16 ... Theorem 1.16 and its proof reads as follows: Question 1 In the second paragraph of above proof...

I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ... I am focused on Section 1.3 Vector Spaces over an Arbitrary Field ... I need help with Exercise 2 of Section 1.3 ... Exercise 2 reads as follows:Hope someone can help with this exercise ... Peter*** EDIT *** To give...

I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ... I am focused on Section 1.2 The Space $$\mathbb{F}^n$$ ... I need help with Exercise 12 ... since I do not get the same answer as the author ... Exercise 12 reads as follows: My attempt at a solution to this apparently...

Homework Statement Find basis and dimension of V,W,V\cap W,V+W where V=\{p\in\mathbb{R_4}(x):p^{'}(0) \wedge p(1)=p(0)=p(-1)\},W=\{p\in\mathbb{R_4}(x):p(1)=0\} Homework Equations -Vector spaces The Attempt at a Solution Could someone give a hint how to get general representation of a vector...

Homework Statement 1) In a vector space V of all real polynomials of third degree or less find basis B such that for arbitrary polynomial p \in V the following applies: [p]_B = \begin{pmatrix} p'(0)\\p'(1)\\p(0)\\p(1)\end{pmatrix} where p' is the derivative of the polynomial p. Homework...

Why are tangent spaces on a general manifold associated to single points on the manifold? I've heard that it has to do with not being able to subtract/ add one point from/to another on a manifold (ignoring the concept of a connection at the moment), but I'm not sure I fully understand this - is...

In Bruce Cooperstein's book: Advanced Linear Algebra, he gives the following example on page 12 in his chapter on vector spaces (Chapter 1) ... ...I am finding it difficult to fully understand this example ... ... Can someone give an example using Cooperstein's construction ... using, for...

The exercise is: (b) describe all the subspaces of D, the space of all 2x2 diagonal matrices. I just would have said I and Z initially, since you can't do much more to simplify a diagonal matrix. The answer given is here, relevant answer is (b): Imgur link: http://i.imgur.com/DKwt8cN.png...

Show that a set of vectors are linearly dependent if and only if anyone of the vectors can be represented as linear combination of the remaining vectors. I don't know these terms. Vectors I know apart from that other terms. Can someone provide some information in any form for solving this...

I'm trying to understand the definition of maps between vector spaces (in normed vector spaces) listed in the following link: http://ocw.mit.edu/courses/mathematics/18-965-geometry-of-manifolds-fall-2004/lecture-notes/lecture3.pdf On the surface, this seems similar to what I expected from the...

Homework Statement How would one determine if a vector space is a subspace of another one? I think that the basis vectors of the subspace should be able to be formed from a linear combination of the basis vectors of the vector space. However, that doesn't seem to be true for this question: Let...

I've just encountered the following theorem : If T is a linear operator in a complex vector space V then if < v , Tv > =0 for all v in V then T=0 But the theorem doesn't hold in real 2-D vector space as the operator could be the operator that rotates any vector by 90 degrees. My question...

∴Homework Statement Let ℝ>0 together with multiplication denote the reals greater than zero, be an abelian group. let (R>0)^n denote the n-fold Cartesian product of R>0 with itself. furthermore, let a ∈ Q and b ∈ (ℝ>0)^n we put a⊗b = (b_1)^a + (b_2)^a + ... + (b_n)^a show that the abelian...

i want a book that smoothly takes me from finite dimensional vector spaces to infinite dimensional vector spaces. Edit: I am doing this as self study, so i would prefer the book to be easy going without an instructor Thanks

Homework Statement I have the operators ##D_{\beta}:V_{\beta}\rightarrow V_{\beta}## ##R_{\beta\alpha 1}: V_{\beta} \otimes V_{\alpha 1} \rightarrow V_{\beta}\otimes V_{\alpha 1}## ##R_{\beta\alpha 2}: V_{\beta} \otimes V_{\alpha 2} \rightarrow V_{\beta}\otimes V_{\alpha 2}## where each...

Hi all. I was hoping I could clarify my understanding on some basic notions of dual spaces. Suppose I have a vector space V along with a basis \lbrace\mathbf{e}_{i}\rbrace, then there is a unique linear map \tilde{e}^{i}: V\rightarrow \mathbb{F} defined by \tilde{e}^{i}(\mathbf{v})=v^{i}...

Good evening everyone, I hope everyone is having a better evening than myself thanks to this homework problem. Let P be the set of positive numbers. For a,b in P, define a+b=a x b; for a in P and a real number r, define r x a= a^r. Show that P is a vector space using ⊕ as addition and (circle...

Hi all, I've been doing some independent study on vector spaces and have moved on to looking at linear operators, in particular those of the form T:V \rightarrow V. I know that the set of linear transformations \mathcal{L}\left( V,V\right) =\lbrace T:V \rightarrow V \vert \text{T is linear}...

I am spending time revising vector spaces. I am using Dummit and Foote: Abstract Algebra (Chapter 11) and also the book Linear Algebra by Stephen Freidberg, Arnold Insel and Lawrence Spence. I am working on Theorem 10 which is a fundamental theorem regarding an isomorphism between the space of...

Hi, I'm having a bit of difficulty with the following definition of a linear mapping between two vector spaces: Suppose we have two n-dimensional vector spaces V and W and a set of linearly independent vectors \mathcal{S} = \lbrace \mathbf{v}_{i}\rbrace_{i=1, \ldots , n} which forms a basis...

Hi all, Just doing a bit of personal study on vector spaces and wanted to clear up my understanding on the following. This is my description of what I'm trying to understand, is it along the right lines? (apologies in advance, I am a physicist, not a pure mathematician, so there are most...

I am reading Chapter 2: Vector Spaces over $$\mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C}$$ of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding Theorem 2.31 (regarding the direct sum of n vector spaces) on pages 61-62. Theorem 2.31 and its accompanying...

I am reading Chapter 2: Vector Spaces over $$\mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C}$$ of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding Theorem 2.27 (First Isomorphism Theorem) on pages 57-58. Theorem 2.27 and its proof read as follows...

I am revising vector spaces and am looking at their quotient spaces in particular ... I am looking at the theory and examples in Schaum's "Linear Algebra" (Fourth Edition) - pages 331-332. Section 10.10 (pages 331-332) defines the cosets of a subspace as follows: Following the above...

[SIZE="4"]Definition/Summary The 1-forms (or covectors or psuedovectors) of a vector space with local basis (dx_1,dx_2,\dots,dx_n) are elements of a vector space with local basis (dx^1,dx^2,\dots,dx^n) The 2-forms are elements of the exterior product space with local basis (dx^1\wedge...

Can someone explain this to me? Thanks! The component in the ith row and jth column of a matrix can be labeled m(i,j). In this sense a matrix is a function of a pair of integers. For what set S is the set of 2 × 2 matrices the same as the set Rs ? Generalize to other size matrices.

Hello all, I have two sets: \[W={\begin{pmatrix} a &2b \\ c-b &b+c-3a \end{pmatrix}|a,b,c\epsilon \mathbb{R}}\] \[V=ax^{2}+bx+c|(a-2b)^{2}=0\]I need to determine if these sets are sub vector spaces and to determine the dim. I think that W is a sub space and dim(W)=3 (am I right?) I don't...

Hello all, I have these two sets (I couldn't use the notation {} in latex, don't know how). V is the set of matrices spanned by these 3 matrices written below. W is a set of 2x3 matrices applying the rule a+e=c+f \[V=span(\begin{pmatrix} 1 &1 &1 \\ 1 &3 &7 \end{pmatrix},\begin{pmatrix} 0 &0...

I'm going through the text "Linear Algebra Done Right" 2nd edition by Axler. Made it to chapter 4 with one problem I'm unable to understand fully. The theory that two vector spaces are isomorphic if and only if they have the same dimension. I can see this easily in one direction, that is...

Hi, can somebody help me with the problem: Suppose that in a vector space over field of real numbers a positive defined norm is defined for each vector which satisfies the triangle inequality and ||aU||=|a|*||u||. Show that a real valued scalar product can de defined as follows...

Here is the question: Here is a link to the question: Let {e1, e2, e3} be a basis for the vector space V and T: V -> V a linear transformation.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Let A and B be subsets of a vector space V. Assume that A ⊂ span(B) and that B ⊂ span(A) Prove that A = B. I don't know how to go about this question, any help would be appreciated.

Hello everyone, I’m looking for very good books of advanced algebra that have a lot of information about vector spaces algebra, in particular. Would you suggest anyone? Many thanks Best regards

I have a few questions about proving that a set is a vector space. 1.) My book lists 8 defining properties of a vector space. I won't list them because I'm under the impression that these are built into the definition of a vector space and thus are common knowledge. My book also says that...

Homework Statement Hi guys , I have this problem ,well actually I don't understand the solution they provide , Here's the problem statement and the solution . May someone please explain the solution to me?? Thanks so much, Sorry for my bad english Homework Equations 1.I understand...

Homework Statement a) The set of real polynomials of x divisible by x^2 + x + 1; b) The set of differentiable functions of x on [0,1] whose derivative is 3x^2 c) all f \in F[0,2] such that x \geq |f(x)| for 0 \leq x \leq 2 The Attempt at a Solution a) Yes, it's a vector space, proven...

Homework Statement In quantum mechanics, what objects are the members of the vector space V? Give an example for the case of quantum mechanics of a member of the adjoint space V' and explain how members of V' enable us to predict the outcomes of experiments. Homework Equations The Attempt...

Author: Paul Halmos Title: Finite-Dimensional Vector Spaces Amazon Link: https://www.amazon.com/dp/0387900934/?tag=pfamazon01-20

Homework Statement Hi there, I'm very new to vector spaces and just can't seem to figure this one problem out. The question ask's to determine if (V,+,*) is a vector space. I am given V=R^2 (x,y)+(x',y')=(x+x'+1,y+y'+1) for addition on V and λ*(x,y)=(λx+λ-1,λy+y-1) (λ∈ℝ) for...

Homework Statement Are the following sets subspaces of R3? The set of all vectors of the form (a,b,c), where 1. a + b + c = 0 2. ab = 0 3. ab = ac Homework Equations Each is its own condition. 1, 2 and 3 do not all apply simultaneously - they're each a separate question. The...

What are some of the strangest vector spaces you know? I don't know many, but I like defining V over R as 1 tuples. Defining vector addition as field multiplication and scalar multiplication as field exponentiation. That one's always cool. Have any cool vector spaces? Maybe ones not over R but...

Homework Statement for the 2x2 matrix [a 12;12 b] is it a vector spaceHomework Equations 1. If u and v are objects in V, then u+v is in V 2. u+v = v + u 3. u+(v+w) = (u+v)+w 4. There is an object 0 in V, called a zero vector for V, such that 0+u = u+0 = u for all u in V 5. For each u in V...

Does a vector in an absract vector space (without any further structure i.e. no inner product or norm) have the properties usually associated with vectors, that is, magnitude and direction? If not, isn't the name vector space a bit misleading and it would be more appropriate to call it a linear...

I'm trying to understand tensor product of vector spaces and how it is done,but looks like nothing that I read,helps! I need to know how can we achieve the tensor product of two vector spaces without getting specific by e.g. assuming finite dimensions or any specific underlying field. Another...

Hello friends, I am looking for an idea to my exercise! let's E be a vector space, e_ {i} be a basis of E, b_ {a} an element of E then b_ {a} = b_ {a} ^ {i} e_ {i}. I want to define a family of vectors {t_ {i}}, that lives on E , (how to choose this family already, it must not be a...

Homework Statement Let F be an infinite field (that is, a field with an infinite number of elements) and let V be a nontrivial vector space over F. Prove that V contains infinitely many vectors. Homework Equations The axioms for fields and vector spaces. The Attempt at a Solution...

I thought that I should do containment in both directions. I have containment in one direction, but the other is much harder. Any ideas?

Theorem : if S ={ v1 , ... , vn} spans the V.Space V , L={w1 , ... , wm} is set of linear independent vectors in V then , n is bigger than or equal to m How can we prove this ? _____________ I read this theorem as a important note but the proof was ommited

Demonstrate with the help of a counter-example why the following is not a vector space. 1. A= ((x,y) \ni R^{2}/ x\geq0) I have many more questions like this, but since I cannot get the first one I think I might have a chance if I understand it. As far as an attempt at an answer, I can...