Vector spaces Definition and 284 Threads

Homework Statement Let V be the vector space of all real-coefficient polynomials with degree strictly less than five. Find the eigenvalues and their geometric multiplicities for the following maps from V to V: a) G(f) = xD(f), where f is an element of V and D is the differentiation map...

Vector Spaces, Polynomials "Over Fields" What does it mean when a vectors space is "over the field of complex numbers"? Does that mean that scalars used to multiply vectors within that vector space come from the set of complex numbers? If so, what does it mean when a polynomial, p(x) is...

I was given a set of problems and answers, but not the solutions. This is one of the questions I've been having trouble in getting the correct answer. Homework Statement Determine whether the set of vectors (u, v) is a vector space, where u + v = 0. Homework Equations The 10 axioms of...

Homework Statement The vector |p> is given by the function x+2x2 and the operator A = 1/x * d/dx, with x = [0,1]. a) Compute the norm of |p> b) Compute A|p>. Does A|p> belong to the VS of all real valued, continuous functions on the interval x = [0,1]? c) Find the eigenvalues and...

Homework Statement Are the following vector spaces over \Re, with the usual notion of addition and scalar multiplication: V = {(0, 1), (1, 0)} Homework Equations definition of vector space The Attempt at a Solution I'm a little confused by what this means. Am I correcting in...

Homework Statement Is the set of all continuous functions (defined on say, the interval (a,b) of the real line) a vector space? Homework Equations None. The Attempt at a Solution I'm inclined to say "yes", since if I have two continuous functions, say, f and g, then their sum f+g...

i'm not sure if I'm posting this in the right place, so forgive me if I'm wrong! in my linear algebra revision i found that I'm struggling with one of the questions: Let S and T be subsets of a vector space, V. Which of the following statements are true? Give a proof or a counterexample. a)...

Homework Statement Proof:Suppose that V is a real inner product space and T\in \wp(V). If (v1... vn) is a basis for V consisting of eigenvectors for T, then there exists an inner product for V such that T is self-adjoint. Homework Equations The Real Spectral Theorem: Suppose that V is a...

Homework Statement Write x=(6,-1,-2)T as x=y+z where y belongs to null A and z belongs to row A A=[1,3,1;2,6,2;-2,-5,0;1,4,3] Homework Equations The main question asks to find all the fundamental subspaces and their dimensions, which I have already found, and then asks me to find the...

Homework Statement Fix a<b in R, and consider the two norms Norm(f)1:=Integralab( Modulus(f)) and Norm(f)Infinity:= sup{Mod(f(x)): a <= x <= b} on the vector space C[a,b]. This question shows that they are not equivalent. a. Show that there is K in R such that for all f in C[a,b]...

Homework Statement Prove that if V is a finite-dimensional vector space, then the space of all linear transformations on V is finite-dimensional, and find its dimension. Homework Equations The Attempt at a Solution

Let U and V be vector spaces over the complex numbers C. Then the tensor product over C, U\otimes_CV is also a complex vector space. Note that U, V, and U\otimes_CV can be regarded as vector spaces over the real numbers R as well. Also note that we can form U\otimes_RV. Question: are U\otimes_CV...

In Rudin's book Functional Analysis, he makes the following claim about the interior A^\circ of a subset A of a topological vector space X: If 0 < |\alpha| \leq 1 for \alpha \in \mathbb C, it follows that \alpha A^\circ = (\alpha A)^\circ, since scalar multiplicaiton (the mapping f_\alpha: X...

I have just started a study of linear algebra and I have a doubt regarding vector spaces. Consider the vector space spanned by the 3 dimensional vectors [1,0,0] and [0,1,0] , this would be a 2-dimensional vector space no doubt.But it also is a subspace of \mathbb R ^3.I have no problem in...

Homework Statement Is U = {f E F([a, b]) | f(a) = f(b)} a subspace of F([a, b]), where F([a, b]) is the vector space of real-valued functions dened on the interval [a, b]? (keep in mind that in the definition of U, the E means belonging to.. I couldn't find an epsilon character)...

Homework Statement Show that the set of all positive real numbers, with x+y and cx redefined to equal the usual xy and xc, is a vector space. What is the zero vector? The Attempt at a Solution My attempt stops at me trying to decipher the problem. Are they asking me to take particular...

Dear all, I'm reading the tensor part of "A course in modern mathematical physics" by Szekeres and I have trouble understanding a concept that you can find in the attached image of the book page. What are the elements of F(V)? If my understanding of (external) direct sums of vector spaces is...

This is an exercise in a linear algebra textbook that I initially thought was going to be easy, but it took me a while to make the proof convincing. Prove: Any subspace of a finite-dimensional vector space is finite-dimensional. Here's my attempt. I am not sure about some details and I'm...

If we could find the Hamel basis for any infinite dimensional vector space, what kind of consequences would this have?

I was working on a problem earlier today and I didn't know the following result: Let S be a subset of an infinite-dimensional vector space V. Then S is a basis for V if and only if for each nonzero vector v in V, there exists unique vectors u1,u2,...,un in S and unique nonzero scalars...

Hello all, Recently I have read something about duality between vector spaces, however my intuition towards this is not clear. Wish someone can give me a hint. Recall the definition of a dual pair is a 3-tuple (X,Y,\langle \cdot , \cdot \rangle) , so essentially duality between vector...

Hello, I'm currently reading Halmos's book "Finite dimensional vector spaces" and I find it excellent. However, I'm having some problems with his definition of the tensor product of two vector spaces, and I hope you could help me clear it out. Here's what he writes: "Definition: The tensor...

Hello, If I am given a vector space (e.g. \mathbb{R}^n), and a group G that acts on \mathbb{R}^n, what are the conditions that G must satisfy so that for any given x\in\mathbb{R}^n its orbit Gx is a manifold ?

Hi: Given a fin.dim vector space V over R, and two different bases B_V,B_V' for V , we say that B_V,B'_V are equivalent as bases ( or have the same orientation) , if there exists a matrix T with TB=B', and DetT>0. How do we define equivalent bases for vector spaces over the...

Homework Statement Determine if the following sets are vector spaces Part A) R^2; u+v = (u1 + v1, u2 + v2); cu = (cu1,cu2) part b) M_2,2 A + B = |(a_11 + b_11) (a_12 + b_12)| & cA = |ca_11 ca_12| |(a_21 + b_21) (a_22 + b_22)| |ca_21...

Homework Statement Homework Equations The 10 axioms: 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 factor for V, such that 0+u = u+0 = u for all u in V 5. For each u in V, there is an object -u in V...

Hello. My question is: does the norm on a space depend on the choice of basis for that space? Here's my line of reasoning: If the set of vectors V = \left\{ v_1,v_2\right\} is a basis for the 2-dimensional vector space X and x \in X, then let \left(x\right)_V = \left( c_1,c_2\right)...

This isn't a homework question, but I thought it'd still be an appropriate place. So this is about vector spaces. I know these are sort of abstract spaces but I'd like more explanation on them. 1) \mathbb{R} This is the space of real vectors right, like from real numbers? 2) \mathbb{P}...

Homework Statement Prove that the following are equivalent: 1. N(A)=0 2. A is nonsingular 3. Ax=b has a unique solution for each b that exists in R^n. Homework Equations The Attempt at a Solution I think you prove this by showing that 1 implies 2, 2 implies 3, & 3 implies 1...

Dear all, I've read the math that defines a tensor product by means of the universal property and I've studied the tensor product construction through a quotient of the free vector space on the cartesian product of two vector spaces. All other constructions of the tensor products are naturally...

Homework Statement For each of the following subsets U of the vector space V decide whether or not U is a subspace of V . Give reasons for your answers. In each case when U is a subspace, find a basis for U and state dim U Homework Equations V=P_{3} ; U=\left\{p\in\...

Hi guys The norm is a continuous function on its vector space, but I a little unsure of how to interpret this. Does it mean that if we are in e.g. a Hilbert space with an orthonormal basis ei (i is a positive integer), then we have \left\| {\mathop {\lim }\limits_{N \to \infty }...

Hey All, I am tutoring a mixed class of (mostly) engineers and physicists and I am trying to get across how important the concept of a vector space is - that its not just some abstract problem that only pure mathematicians need to worry about. Its easy to highlight the need for linear...

I still don't fully understand the explicit construction of the tensor product space of two vector spaces, in spite of the efforts by several competent posters in another thread about 1.5 years ago. I'm hoping someone can provide the missing pieces. First, a summary of the things I think do...

Homework Statement 1) Determine if a) (a,b,c), where b=a+c b) (a,b,0) are subspaces of R3 and 2) Determine whether the given vectors span R3 a) v1 = (3,1,4) v2 = (2,-3,5) v3 = (5,-2,9) v4 = (1,4,-1) Homework Equations - If u and v are vectors in W, then u + v is in W -...

Homework Statement T = {(1,1,1),(0,0,1)} is a subset of R^{3} but not a subspace sol i have to prove it holds for addition and scalar multiplication so let x=(1,1,1) and y =(0,0,1) so x+y = (1,1,2) so it holds let \alpha = a scalar then \alphax = (\alpha,\alpha,\alpha)...

Homework Statement First of all sorry if my terminology sounds a bit weird, i have never studied mathematics in english before. So this is the problem: We have the space R^2x2 of all the tables with numbers in R. We also have a subspace V of R^2x2 of all the tables with the following...

I currently self study from the book "A Course in Modern Mathematical Physics" by Peter Szekeres, and I'm currently reading the chapter on tensors, which he defines using the concept of Free Vector Spaces. He gives a re-definition of Grassmann's algebras introduced in the previous section by...

Homework Statement Prove that if V and W are three dimensional subspaces of R5, then V and W must have a nonzero vector in common. Homework Equations NA The Attempt at a Solution I've attempted to set up the problem by writing out, V = { (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0...

Homework Statement part a. Use the matrix A = {[1,-1,0] [0,-1,1] [-1,2,-1]} to compute T(x) for x = {[1] [2] [3]} Here, T:R^3-->R^3 is defined as T(x)=Ax. part b. describe the kernel of the transformation. part c. what is the nullity of the tarnsformation part d. what...

The linear algebra course I'm taking just became very "wordy" and I am having a hard time dealing notions such as subspaces without a diagram. I was thinking Venn diagrams could be used to visualize relationships between subspaces of vector spaces. Has this been a useful way to organize the...

Massively stuck with this one, have done some reading and am having difficulty connecting matrices to vector spaces (a) Verify that the space of the real (2 x 2)-matrices, endowed with the standard addition and multiplication by real scalars, forms a vector space (b) Specify a basis for...

Hi everyone, I would need to get some help on the following question Let A (m*n) Let B (m*p) Let L(A) be the span of the columns of A. L(A) is orthogonal to L(B) <=> A'B=0 I suppose that the => direction is pretty obvious, since A is in L(A) and B in is L(B). Now I am not sure how to...

On pg. 2, Shankar seems to just assume (I'm guessing) that 1|V\rangle = |V\rangle for all vectors |V\rangle when he does the exercise at the bottom of the page. Is this true, or is it possible to prove 1|V\rangle = |V\rangle from the axioms he lists?

This subject came up in some notes on linear algebra I'm reading and I don't get it. Please help me understand. -- First, the relevant background and notation relating to my question: Let S be a nonempty set and F be a field. Denote by l(S) the family of all F-valued functions on S and...

Can someone explain how topological vector spaces are more general than normed ones. So this means that if I have a normed vector space, X, it would have to induce a topology. I was thinking the base for this topology would consist of sets of the form \{y\in X: ||x-y||<\epsilon, \textrm{for some...

Alrighty then :smile: I am working through Axler's LA Done Right. I have 2 questions for now: 1.) He uses some notation that is confusing me. When referring to \mathbf{R} \text{ or }\mathbf{C} as a set it is in BOLD but when he refers to a vector space as being the set V, it is not in...

Why are they called "axioms"? Shouldn't they be called "definitions"?

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...