Vector spaces Definition and 285 Threads

What is the best way of describing isomorphism between two vector spaces? Is there a real life analogy of isomorphism?

What happen if i drop the "V is a vector spaces. If k,l \in \mathbb{F}, u \in V, (k+l)u=ku + lu" rule in vector spaces?

Homework Statement Hi, I am really having trouble with questions regarding proving whether a given set is a vector space or not. So one of the questions is [ x ε R2|x12=x23 ] So I have to prove whether the following set is a vector space Homework Equations The Attempt at a...

According to my professor, there exist infinite dimensional vector spaces without a basis, and he asked us to find one. But isn't this impossible? The definition of a dimension is the number of elements in the basis of the vector space. So if the space is infinite-dimensional, then the basis...

Homework Statement This question came out of a section on Correspondence and Isomorphism Theorems Let V be a vector space and U \neq V, \left\{ \vec{0} \right\} be a subspace of V. Assume T \in L(V,V) satisfies the following: a) T(\vec{u} ) = \vec{u} for all \vec{u} \in U b) T(\vec{v} + U) =...

Hello, I am wondering why is it that matrices and infinite sequences may be considered part of a vector space. I have read 3 different sources, and my interpretation of a vector space is something that belongs in a field and follows a list of properties that are standard to real numbers, i.e...

If A and B are vector spaces over ℝ or ℂ show that a sequence (a_n, b_n) in A×B converges to (a,b) in A×B only if a_n converges to a in A and b_n converges to b in B as n tends to infinity. To me this statement sounds pretty intuitive but I have been having trouble actually proving it...

Homework Statement Let V be a vector space over the field F and consider F to be a vector space over F in dimension one. Let f \in L(V,F), f \neq \vec{0}_{V\rightarrow F}. Prove that V/Ker(f) is isomorphic to F as a vector space. Homework Equations L(V,F) is the set of all linear maps...

In 'Linear Algebra Done Right' by Sheldon Axler, a direct sum is defined the following way, We say that V is the direct sum of subspaces U_1, \dotsc ,U_m written V = U_1 \oplus \dotsc \oplus U_m, if each element of V can be written uniquely as a sum u_1 + \dotsc + u_m, where each u_j \in U_j...

Hi, I have this problem that is solved, but I don't understand the theory behind it. It says: Which of the following sets, with the natural definitions of addition and scalar multiplication, form real vector spaces? A) The set of all differentiable functions f:(0,1)\rightarrow\Re such that...

Homework Statement let S={(a_1,a_2):a_1,a_2 \in \mathbb{R}} For (a_1,a_2),(b_1,b_2)\in{S} and c\in\mathbb{R} define (a_1,a_2)+(b_1,b_2)=(a_1+b_1,a_2-b_2) and c(a_1,a_2)=(ca_1,ca_2). show that this is not a vector space Homework Equations vector space axioms The Attempt at a...

Students familiar with Euclidean space find the introduction of general vectors spaces pretty boring and abstract particularly when describing vector spaces such as set of polynomials or set of continuous functions. Is there a tangible way to introduce this? Are there examples which will have a...

Students familiar with Euclidean space find the introduction of general vectors spaces pretty boring and abstract particularly when describing vector spaces such as set of polynomials or set of continuous functions. Is there a tangible way to introduce this? Are there examples which will have a...

Homework Statement If (V,\omega) is a symplectic vector space and Y is a linear subspace with \dim Y = \frac12 \dim V show that Y is Lagrangian; that is, show that Y = Y^\omega where Y^\omega is the symplectic complement. The Attempt at a Solution This is driving me crazy since I...

Homework Statement Does the vector space of all square matrices have a basis of invertible matrices? Homework Equations No relevant equations. The Attempt at a Solution I know that the 2x2 case has an invertible basis, but I don't know how to generalize it for the vector space...

Homework Statement Homework Equations From my notes I'm aware of the following equation: dim(A + B) = dimA + dimB − dim(A ∩ B).The Attempt at a Solution I'm assuming part of the solution involves the equation above and rearranging it but I'm not sure how I would determine dim(A + B). I also...

I have a question concerning subspaces of infinite dimensional vector spaces. Specifically given any infinite dimensional vector space V, how might one construct an infinite decreasing chain of subspaces? That is: V=V0\supseteqV1\supseteq... , where each Vi is properly contained in Vi-1...

Hi, I have read my notes and understand the theory, but I am having trouble understanding the following questions which are already solved (I am giving the answers as well). The first question says: Let U_{1} and U_{2} be subspaces of a vector space V. Give an example (say in V=\Re^{2}) to...

Homework Statement Hello, I'm having a little difficulty with this proof. Prove: If V is a complex vector space, then V is also a real vector space. Homework Equations Definition 1: A vector space V is called a real vector space if the scalars are real numbers. Definition 2: A vector...

1. Homework Statement Prove that there is an additive identity 0∈R^n: For all v∈R^n, v+0=v2. Homework Equations Axiom of Real Numbers: There is an additive identity 0∈R : For all a∈R, a+0=a and o+a=a 3. The Attempt at a Solution Solution 1 (My own attempt) : Let v=(v1, v2, v3... vn). Then...

Homework Statement For the vector set<a1,a2>, where (a1a2 < equal to 0) Homework Equations The Attempt at a Solution I'm not sure why this set is close under scalar multiplication and not in vector addition. Some hints would be nice :D

In studying vector spaces, I came across the coset of a vector space. We have an equivalence relation defined as u = v \rightarrow u-v \in W where W is a subspace of V. the equivalence class that u belongs to is u + W. I can see why u must belong to this equivalence class ( the...

Hello, I am struggling with this question... U is a set of all matrices of order 3X3, in which there is at least one row of 0's. W is the set of matrices: a b c b+c-3a where a,b,c are real numbers. V is the set of vectors: (x,y,z,w), for which 5(y-1)=z-5 which two of these statements...

Suppose that F/K is an algebraic extension, S is a subset of F with S/K a vector space and s^n \in S for all s in S. I want to show that if char(K) isn't 2, then S is a subfield of F. Since F/K is algebraic, we know that \text{span} \lbrace 1,s,s^2,...\rbrace is a field for any s in S...

Hi, Everyone: Just curious about two things: 1) if we are given the complexes as a vector space V over R , so that z1,..,zn are a basis, I heard there is a "natural way" of turning this into a vector space of R^{2*n} over R; IIRC , is this how it is done: {...

When reading in Griffiths and on Wikipedia about the vector space formulation of wavefunctions, i am constantly faced with the statement that a vector can be expressed in different bases, but that it's still the same vector. However, I'm having a hard time imagining what it is about a vector...

Homework Statement The set of all pairs of real numbers of the form (1,x) with the operations: (1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx) k being a scalar Is this a vector space?Homework Equations (1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx)The Attempt at a Solution I verified most of the axioms...

Finite-dimensional V and W are linearly isomorphic vector spaces over a field. Prove that if \{v_{1},...,v_{n}\} is a basis for V, \{T(v_{1}),...,T(v_{n})\} is a basis for W. My attempt at a proof: Let T:V\rightarrow W be an isomorphism and \{v_{1},...,v_{n}\} be a basis for V. Since T is an...

Are There "Unnormed" Vector Spaces? Apologies if this question is barking up a ridiculous tree, but: as I understand it, a normed vector space is simply a vector space with a norm. This seems to suggest the existence of vector spaces without norms. My question is whether these are vector spaces...

Decide if the indicated set of functions are independent or dependent, and prove your answer. \left\{cos^2(x),sin^2(x),sin(2x)\right\} This linear algebra course is killing me. It's much more abstract than I thought it would be. I realize this problem isn't exactly that, but I am so...

Let K1={a + (2^0.5)*b} | a,b rational numbers}, and K2={a + (3^0.5)*b} | a,b rational numbers} be two fields with the common multiplication and addition. Isomorphs are the following vector spaces : (Q^n ., +; K1) and (Q^n ., +; K2) ?

Hey guys So the problem I'm having here is spans! I know that a basis of a vector space is a linearly independent spanning set So the linearly independent part is pretty easy...looking at whether or not vectors are linear combinations of the others (right? Or do I have to look at it by...

Homework Statement Homework Equations addition / multiplication -- showing that the vectors (in this case) aren't closed under addition and scalar multiplication. . . The Attempt at a Solution I really get stuck with proofs: how to begin / where to go / etc. I know that I am...

Homework Statement V is the set of functions R -> R; pointwise addition and (a.f)(x) = f(ax) for all x. is V a vector space given the operations?Homework Equations nil. The Attempt at a Solution i think it is not closed under multiplication. if r is an element of R, then r*a(x) . r*f(x)...

Homework Statement Given a Hermitian operator A = \sum \left|a\right\rangle a \left\langle a\right| and B any operator (in general, not Hermitian) such that \left[A,B\right] = \lambdaB show that B\left|a\right\rangle = const. \left|a\right\rangle Homework Equations Basically those...

Homework Statement Show whether the set is a vector space: The set of all triples of real numbers (x, y, z) with the operations: (x, y, z) + (x', y', z') = (x + x', y + y', z + z') and k(x, y, z) = (kx, y, z) Homework Equations (10 vector space axioms) The Attempt at a Solution...

Homework Statement [PLAIN]http://i26.lulzimg.com/274748.jpg Homework Equations ?? The Attempt at a Solution i don't even know how to start. lol.

Hi, I have a problem understanding the difference between Cartesian product of vector spaces and tensor product. Let V1 and V2 be vector spaces. V1 x V2 is Cartesian product and V1 xc V2 is tensor product (xc for x circled). How many dimensions are in V1 x V2 vs V1 xc V2? Thanks, Monte

Hi, I'm new to linear algebra. I'm pretty good at doing exercises with matrices and stuff but even though I've been looking in different books, looking all over the internet I can't get into vector spaces and subspaces. It seems like the books have some very elementary and simple examples and...

Homework Statement Let R be a commutative ring, and let F = R^{\oplus B} be a free R-module over R. Let m be a maximal ideal of R and take k = R/m to be the quotient field. Show that F/mF \cong k^{\oplus B} as k-vector spaces. The Attempt at a Solution If we remove the F and k...

I first introduce the vector along the lines 'something with magnitude and direction'. Later on the definition of a vector becomes generic - 'an element of a vector space'. Euclidean spaces (n=2 and n=3) are something we can all visualize. However when describing other vector spaces such as...

I was hoping you guys could help me in understanding some vector spaces of infinite dimension. My professor briefloy touched n them (class on linear algebra), but moved on rather quickly since they are not our primary focus. He gave me the example of the closed unit interval where f(x) is...

Homework Statement Show if V is a vector space ([a,b,c]|ab>=0). I'm trying to test whether it is closed under vector addition. Homework Equations v=[a1,b1,c1] w=[a2,b2,c2], v and w satisfy ab>=0 a1b1>=0, a2b2>=0 show (a1+a2)(b1+b2)>=0 The Attempt at a Solution Got to a1b1...

If you find the exact same basis for two vector spaces, then is it true that the vector spaces are equal to each other?

Homework Statement Does ||x||_inf = max | x_i | for 1 <= i <= n define a norm on R^(n) Homework Equations The Attempt at a Solution ok, I thought I understood vector spaces but this problem is confusing the heck out of me. A norm is a function that assigns a positive and finite length to...

1) How to show that if W is a subspace of a finite-dimensional vector space V, then W is finite-dimensional and dim W<= dimV. 2) How to show that if a subspace of a finite-dimensional vector space V and dim W = dimV, then W = V. 3) How to prove that the subspace of R^3 are{0}, R^3 itself...

Homework Statement Prove that the set of all scalar multiples of the vector [1,3,2] in R3 forms a vector space with th usual operations on 3-vectors.Homework Equations I am struggling to get anywhere on with this on paper. I know intuitively it and since its an intro course its a elementary...

hi.. can anyone say what is the concept behind change of basis.. y do we change a vector of one basis to another?

Homework Statement I have quite a few problems that I believe I answered correctly, but here is one of them: 1. Rather than use the standard definitions of addition and scalar multiplication in R3, suppose these two operations are defined as follows: (x1, y1, z1) + (x2, y2, z2) = (x1+ x2...

Homework Statement Just started learning vector spaces... not as fun as matrices. Anyway, I have a problem here, and I just want to make sure I'm understanding it correctly. It states: "The set {(x,y): x>/0; y>/0} with the standard operations in R^2." It asks me to prove whether or not it's a...