Subspace Definition and 560 Threads

http://imageshack.us/a/img141/4963/92113198.jpg hey, I'm having some trouble with this question, For part a) I know that in order for c_0 to be closed every sequence in c_0 must converge to a limit in c_0 but I am having trouble actually showing that formally with the use of the norm...

Homework Statement For F \in {R,C} and for an infinitie discrete time-domain T, show that lp(T;F) is a strict subspace of c0(T;F) for each p \in [1,∞). Does there exist f \in c0(T;F) such that f \notin lp(T;F) for every p \in [1,∞) Homework Equations Well we know from class that...

The problem has been attached. I am having difficulty expressing myself. The professor said for this problem, it would be best if I use words to answer it. 1. I must verify the 0 vector is in S+T. Since S and T are subspaces, the 0 vector must exist in both S and T. Thus 0+0=0 and 0 vector is...

I have attached the problem. part a) show that S is a subspace of R4 I have to show the following 3 conditions 0 vector is in S if U and V are in S, then U+V is in S If V is in S, then cV is in S where c is a scalar if s and t=0 which are real #s, then the 0 vector is in R4, thus S...

Homework Statement Consider the vector space F(R) = {f | f : R → R}, with the standard operations. Recall that the zero of F(R) is the function that has the value 0 for all x ∈ R: Let U = {f ∈ F(R) | f(1) = f(−1)} be the subspace of functions which have the same value at x = −1 and x = 1...

1. Let V={(X^2+X+1)p(x) : p(x) \in P2(x)} Show that V is a subspace of P4(x). Display a basis, with a proof. What is the dimension of V? 2. 3. I started to try to figure out how to prove that V is a subspace of P4, but I'm not sure how. To show that it is closed under addition: p(x)=x^2 is in...

1. Let V={X \in R2 : (1 2) X= (0)} ......(3 4) ... (0) Show that V is a subspace of R^2 with the usual operations. What is the dimension of V 2. Homework Equations 3. I am really kind of lost, the statement seems to make no sense. X is in R but it also = the matrix [1 2, 3 4] but...

Hi, A quick question: Does a set need to be a subset to be a subspace of some vector space?

Homework Statement show that (U\bot)\bot=U, if (ℝ,V,+,[,.,]) an Euclidian space en U is a linear subspace of V. Homework Equations The Attempt at a Solution suppose \beta={u_1,...,u_k} is an orthonormal basis of U. pick u in U. Then u=x_1u_1+...+x_ku_k for certain x_1,...,x_k...

Homework Statement we have the vector space (ℝ,ℝ^N,+) of all sequences in ℝ. if A={(x_n) \in ℝ^N | only finitely many components x_j differ from 0}, show that A is a linear subspace of ℝ^N. With which other vector space is this subspace isomorphous? Homework Equations The Attempt at...

How do i go about this? Find a basis for the subspace W of R^5 given by... W = {x E R^5 : x . a = x . b = x . c = 0}, where a = (1, 0, 2, -1, -1), b = (2, 1, 1, 1, 0) and c = (4, 3, -1, 5, 2). Determine the dimension of W. (as usual, "x . a" denotes the dot (inner) product of the...

Find a set of vectors in \mathbb{R}^3 that spans the subspace S\,=\,\{\,u\,\in\,\mathbb{R}^3\,|\,u\cdot v\,=\,0\,\} where v=<1,2,3> Maybe 12 hours of studying is too much and I'm fried or, maybe I'm looking for excuses. Either way... To solve this I'm trying to set up a matrix...

Homework Statement Let S = { p \in P_{2}(ℝ) | p(7) = 0 } Prove that S is a subspace of P_{2}(ℝ) (the vector space of all polynomials of degree at most 2) The Attempt at a Solution So essentially I have to prove that S is closed under addition, scalar multiplication and that...

1. Hello, I'm reading through Munkres and I was doing this problem. 16.8) If L is a straight line in the plane, describe the topology L inherits as a subspace of ℝl × ℝ and as a subspace of ℝl × ℝl (where ℝl is the lower limit topology). Homework Equations The Attempt at a Solution I've...

1. Homework Statement My question is if u+v is in the subspace can you say that u or v is in the subspace? If not would there be a counterexample? 2. Homework Equations closed under addition/scalar multiplication 3. The Attempt at a Solution I know that if u or v were in the subspace...

Find a basis of the given span{[2,1,0,-1],[-1,1,1,1],[2,7,4,5]} So I got the RREF, and found the basis to be two rows of the RREF, which are [1,0,-1/3,0] and [0,1,2/3,1], but the answer is [2,1,0,-1],[-1,1,1,1] Where did I do wrong?

I'm looking at Rao: Topology, Proposition 1.5.4, "A closed subspace of a Lindelöf space is Lindelöf." He gives a proof, which seems clear enough, using the idea that for each open cover of the subspace, there is an open cover of the superspace. But I can't yet see where he uses the fact that the...

Homework Statement Suppose that (X,\tau) is the co-finite topological space on X. I : Suppose A is a finite subset of X, show that (A,\tau) is discrete topological space on A. II : Suppose A is an infinite subset of X, show that (A,\tau) inherits co-finite topology from (X,\tau). The...

Homework Statement Prove: If ##X## is a (possibly infinite dimensional) locally convex space, ##L \leq X##, ##dimL < \infty ##, and ##M \leq X ## then ##L + M## is closed. Homework Equations The Attempt at a Solution ##dimL < \infty \implies L## is closed in ##X## ##L+M = \{ x+y : x\in L, y...

Homework Statement Let W = \begin{cases} \begin{pmatrix}x\\y\\z\\w\end{pmatrix} \in R^4 | w + 2x + 2y + 4z = 0 \end{cases} A)Find basis for W. B)Find basis for W^{\perp} C)Use parts (A) and (B) to find an orthogonal basis for R^4 with respect to the Euclidean inner product. Homework...

Let W = \{ax^3 + bx^2 + cx + d : b + c + d = 0\} and P_3 be the set of all polynomials of degrees 3 or less. So say we want to prove the W is a subspace of P_3. We let p(x) = a_1x^3 + a_2x^2 + a_3x + a_4 and g(x) = b_1x^3 + b_2x^2 + b_3x + b_4. So, we compute f(x) + kg(x) and the answer...

I've been working on this Linear Algebra problem for a while: Let F be a field, V a vector space over F with basis \mathcal{B}=\{b_i\mid i\in I\}. Let S be a subspace of V, and let \{B_1, \dotsc, B_k\} be a partition of \mathcal{B}. Suppose that S\cap \langle B_i\rangle\neq \{0\} for all i...

I am wondering how to organise all of those concepts in my head. should i think of it like: subspace > vectorspace > nullspace, columnspace kind of like columnspaces and nullspaces are valid vectorspaces, and all of those are valid subspaces. is a vector space a columnspace? except its...

I'm having trouble conceptualizing exactly what a subspace is and how to identify subspaces from vector spaces. I know that the definition of a subspace is: A subset W of a vector space V over a field \textbf{F} is a subspace if W is also a vector space over \textbf{F} w/ the operations of...

εHomework Statement Let W = R3 be the set W = {(x,y,z)|z=y-2x}. Show that W is a subspace of R3 Homework Equations From the equations my teacher gave me I know that if W is to be a subspace it needs to follow: 1.) U,VεW then (U + V)εW, 2.)UεW and K(random constant)εW then KUεW...

Homework Statement Let W = {p(t) ∈ P4(t): p(0)=0 }. Prove that W is a subspace of P4(t)Homework Equations noneThe Attempt at a Solution I know three things have to be true in order to be a subspace: 1. zero vector must exist as an element 2. if u and v are elements, u+v must be an element 3. if...

Homework Statement Hey, I'm trying to figure out whether A = [x+1,0] is a subspace... I know it's probably simple but what I'm confused is that...Homework Equations The Attempt at a Solution u+v must be an element of A. let u = [x1+1,0] and v=[x2+1,0]. Adding them together gives you...

I'm a little confused about some of the matrix terminology. I have the following subspace: span{v1, v2, v3} where v1, v2, v3 are column vectors defined as: v1 = [1 2 3] v2 = [4 5 6] v3 = [5 7 9] (pretend they are column vectors) How am I supposed to find the dimension of the span? My...

I'm having a hard time understanding subspaces and subsets in linear algebra. So what I'm getting is that a subset is any set of vectors in a plane R^n. So a subspace is a set of vectors in a subspace? http://postimage.org/image/6e2tl1c51/ http://postimage.org/image/6e2tl1c51/ ^This is my...

Homework Statement Determine if the set is a subspace: S = {(a, b, c) 2 R3 | a − 2b = 0 and 2a + b + c = 0}; Homework Equations as above The Attempt at a Solution It is a subspace, I'm just not 100% sure how to write up the proof. So far I have this: The set is non-empty...

Hi! I'm trying to work through a script on Riemannian submersions but I have some problems with one proof in particular (or more likely the general underlying concepts). No worries, this is not about the entire proof, but just one step. This is about the quaternionic projective space...

Help: Sound created over/in subspace? (SciFi) I know nothing, called basic high school physics and chemistry. However, do not hold back in your answers I do find ways to comprehend what people tell me. This might be a silly question but since audible "sounds" can't travel over empty...

Find the dimension of the subspace of M2;2 consisting of all 2 by 2 matrices whose diagonal entries are zero. ? i know that the dimension is the number of vectors that are the basis for this subspace ,but i cannot figure out what is the basis for this subspace ? any help will be...

Homework Statement Is this a subspace of R^4, (2x+3y, x, 0 , 1) . Give reasons Homework Equations The Attempt at a Solution I am completely stuck at this one

Hi, All: I have been tutoring linear algebra, and my student does not seem to be able to understand a solution I proposed ( of course, I may be wrong, and/or explaining poorly). I'm hoping someone can suggest a better explanation and/or a different solution to this problem...

Homework Statement Show that every topological manifold is homeomorphic to some subspace of E^n, i.e., n-dimensional Euclidean space. Homework Equations A topological manifold is a Hausdorff space that are locally Euclidean, i.e., there's an n such that for each x, there's a neighborhood...

I've been struggling with this problem for about two weeks. My supervisor is also stumped - though the problem is easy to state, I don't know the proper approach to tackle it. I'd appreciate any hints or advice. Let V be an random k-dimensional vector subspace of ℝn, chosen uniformly over...

Homework Statement Prove if set A is a subspace of R4, A = {[x, 0, y, -5x], x,y E ℝ} Homework Equations The Attempt at a Solution Now I know for it to be in subspace it needs to satisfy 3 conditions which are: 1) zero vector is in A 2) for each vector u in A and each vector v in A, u+v is also...

Hi all I hope you guys can help me. I am soo confused with this question: I would really liked a complete answer to this, I have an upcoming exam and I know these two will be on the exam. 1. Let V be the set of all diagonal 2x2 matrices i.e. V = {[a 0; 0 b] | a, b are real numbers} with...

Homework Statement A is a 2 x 2 matrix. Prove that the set W = {X: XA = AX} is a subspace of M2,2Homework Equations The Attempt at a Solution I have already proven non-emptiness and vector addition. Non-emptiness: W must be non-empty because the identity matrix I is an element of W. IA = AI A...

Vector space V = Rn, S = {(x,2x,3x,...,nx) | x is a real number} I know that it is closed under addition, but to be closed under scalar mult... say r is a scalar, then Say we multiply S by -4 Then -4S = (-4x,-8x,...-4nx) Because the problem never stated that the number infront of x...

Homework Statement Let S be a linearly independent subset of a Hilbert space. Prove that span(S) is a subspace, that is a linear manifold and a closed set, if and only if S is finite. Homework Equations The Attempt at a Solution Assuming S is finite means that S is a closed set...

Homework Statement V is a subspace of the vector space R^{3} given by: V = {(x, y, z) E R^{3} / x + 2y + z = 0 and -x + 3y + 2z = 0} Find a subspace W of R^{3} such that R^{3} = V\oplusW I'm really lost in this. My teacher didn't give any example of how to solve this kind of exercise...

Homework Statement Let [a,b] be a closed, bounded interval of real numbers and consider L^{\infty}[a,b]. Let X be the subspace of L^{\infty}[a,b] comprising those equivalence classes that contain a continuous function. Show that such an equivalence class contains exactly one continuous...

Is an open normed subspace Y (subset of X) primarily defined as a set {y in X : Norm(y) < r}? Where r is some real (positive) number. I know the open ball definitions and such... but it seems like this definition is saying, an open normed space, is essentially an open ball which satisfies...

Let T: V-->W be a surjective linear transformation and let X be a subspace of W. Assume tat Ker(T) and X are finite dimensonal. Prove that T^-1 (X) = {v | T(v) is in X} is a subspace of V. Ok I absolutely suck at showing things are a subspace of something...I don't even know where to...

Homework Statement Suppose A is a fixed matrix in M. Apply the subspace theorem to show that S = {x \in ℝ : Ax = 0}The Attempt at a Solution Zero vector for x = <0,0,0> A*<0,0,0> = 0 Therefore zero vector is in ℝ and S is non-empty. Addition: For u & v \in S u + v = 0 + 0 = 0 A*(u+v) =...

Homework Statement http://img21.imageshack.us/img21/4580/screenshot20120117at218.png The Attempt at a Solutiona) Suppose we have two arbitrary vectors of E, call them X,Y. Let X = (2x,x) where x is in R and let Y = (2y,y) where y is in R. If we add X and Y we have (2x,x) + (2y,y) =...

Homework Statement http://img854.imageshack.us/img854/5683/screenshot20120116at401.png The Attempt at a SolutionSo we have that A + B is a vector in S + T, where A is an element of S and B is an element of T. Suppose there is another vector A' + B' also in S + T, where A' is an element of S...

Homework Statement Let U be the subspace of P3(ℝ) spanned by E={x^3,x^3-x^2,x^3+x^2,x^3-1} find a linearly independent subset F of E spanning U. Homework Equations E={x^3,x^3-x^2,x^3+x^2,x^3-1} The Attempt at a Solution a(x^3)+b(x^3-x^2)+c(x^3+x^2)+d(x^3-1)=0x^3+0x^2+0x+0...