What is Subspace: Definition and 571 Discussions

In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces.

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  1. G

    Show subspace of normed vector is closed under sup norm.

    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...
  2. L

    Prove l^p strict subspace of c0

    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...
  3. C

    Verifying Subspace Properties of S+T

    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...
  4. C

    I have attached the problem.part a) show that S is a subspace of

    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...
  5. 1

    Polynomial Span and Subspace - Linear Algebra

    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...
  6. N

    Proving that V is a Subspace of P4(x) and Calculating its Dimension

    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...
  7. N

    Show Proving V is a Subspace of R2, Dimension of V

    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...
  8. M

    Subspace vs Subset: Understanding the Relationship

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

    Euclidian space en U is a linear subspace

    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...
  10. D

    Is A a Linear Subspace of ℝ^N?

    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...
  11. C

    Basis of a subspace and dimension question

    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...
  12. M

    Find a set of vectors that spans the subspace

    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...
  13. N

    Proving S is a Subspace of P2: A Scientific Approach

    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...
  14. S

    Subspace Topology of a Straight Line

    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...
  15. S

    U+v in subspace W, is u or v in subspace

    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...
  16. G

    Find a basis of a subspace of R^4

    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?
  17. Rasalhague

    Closed subspace of a Lindelöf space is Lindelöf

    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...
  18. A

    [Topology] Find the open sets in the subspace topology

    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...
  19. C

    Prove: sum of a finite dim. subspace with a subspace is closed

    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...
  20. H

    Orthogonal Basis for a subspace

    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...
  21. H

    Proving W is a Subspace of P_3 & Finding a Vector Not in W

    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...
  22. I

    Is a subspace the direct sum of all its intersections with a partition of the basis?

    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...
  23. V

    Subspace, vectorspace, nullspace, columnspace

    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...
  24. K

    What Are the Requirements for a Subset to Be Considered a Subspace?

    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...
  25. J

    Show that W is a subspace of R3

    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...
  26. B

    Prove that W is a subspace of P_4(t)

    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...
  27. I

    Quick Subspace Question: Understanding A = [x+1,0] as a Potential Subspace

    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...
  28. D

    Dimension of a subspace question

    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...
  29. G

    What is a subspace and a subset and how are they related?

    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...
  30. A

    Show that the set S is a subspace of R^3

    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...
  31. S

    Vertical subspace in HPn (Riemannian submersions)

    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...
  32. S

    Help: Sound created over/in subspace? (SciFi)

    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...
  33. B

    What is the Dimension of the Subspace of M2;2 with Zero Diagonal Entries?

    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...
  34. S

    Subspace in R^4: Investigating (2x+3y, x, 0, 1) as a Potential Subspace

    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
  35. B

    Find T with Subspace S as Kernel of T

    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...
  36. S

    Prove that topological manifold homeomorphic to Euclidean subspace

    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...
  37. P

    Dimension of an intersection between a random subspace and a fixed subspace

    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...
  38. I

    Prove if set A is a subspace of R4

    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...
  39. M

    Is V a vector space? and Show that W is not a subspace

    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...
  40. S

    Prove that the set {X: XA = AX} is a subspace of M 2,2

    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...
  41. M

    Closed Under Scalar Multiplication: Subspace Question Vector Space V = Rn

    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...
  42. W

    Span of a linearly independent subset of a hilbert space is a subspace iff finite

    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...
  43. C

    Linear Algebra - Find a subspace

    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...
  44. J

    Proving C[a,b] is a Closed Subspace of L^{\infty}[a,b]

    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...
  45. B

    Functional Analysis: Open normed subspace

    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...
  46. T

    Prove the inverse image is a subspace.

    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...
  47. N

    Subspace theorem problem (matrix)

    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) =...
  48. T

    Identifying Subspaces and Subspace Addition

    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) =...
  49. T

    Proving Uniqueness in Subspace Addition

    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...
  50. T

    Proving Uniqueness in Subspace Addition

    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...
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