What is Subspace: Definition and 570 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. M

    Vector Subspaces: Determining U as a Subspace of M4x4 Matrices

    Determine whether the following subsets U of M4x4is a subspace of the vector space V of all M4x4 matrices, with the standard operations of matrix addition and scalar multiplication. If is not a subspace provide an example to demonstrate a property that U does not possess. a. The set U of all 4x4...
  2. S

    I Smallest subspace if a plane and a line are passing through the origin

    Hi all, I am a beginner in Linear Algebra. I am solving problems on vector spaces and subspaces from the book Introduction to Linear Algebra by Gilbert Strang. I have come across the following question: Suppose P is a plane through (0,0,0) and L is a line through (0,0,0). The smallest vector...
  3. P

    Prove that ##S## is a subspace of ##V##

    Let ##S## be the subset of real (infinite) sequences (##a_1,a_2,\ldots##) with ##\lim a_n=0## and let ##V## be the space of all real sequences. Is ##S## a subspace of ##V##? Hello. I want to ask for help to start solving this problem. I don't understand how I can apply the theory I've studied...
  4. A

    Engineering Signals & Systems with Linear Algebra

    Hello everyone, I would like to get some help with the above problem on signals and linear projections. Is my approach reasonable? If it is incorrect, please help. Thanks! My approach is that s3(t) ad s4(t) are both linear combinations of s1(t) and s2(t), so we need an orthonormal basis for the...
  5. H

    I Can this work as a basis for S?

    Let ##S## be a set of all polynomials of degree equal to or less than ##n## (n is fixed) and ##p(0)=p(1)##. Then, a sample element of ##S## would look like: $$ p(t) = c_0 + c_1t +c_2t^2 + \cdots + c_nt^n $$ Now, to satisfy ##p(0)=p(1)## we must have $$ \sum_{i=1}^{n} c_i =0 $$ What could...
  6. J

    MHB Basis of linear subspace

    I have a given point (vector) P in R^3 and a 2-dimensional linear subspace S (a plane) which consists of all elements of R^3 orthogonal to P. The point P itself is element of S. So I can write P' ( x - P ) = 0 to characterize all such points x in R^3 orthogonal to P. P' means the transpose...
  7. Norashii

    Proof of Subspace Topology Problem: Error Identification & Explanation

    I have already seen proofs of this problem, but none of them match the one I did, therefore I would be glad if someone could indicate where is the mistake here. Thanks in advance. **My proof:** Take a limit point x of U that is not in U, but is in K (in other words x \in K...
  8. JD_PM

    Finding a complementary subspace ##U## | Linear Algebra

    We only worry about finite vector spaces here. I have been taught that a subspace ##W## of a vector space ##V## has a complementary subspace ##U## if ##V = U \oplus W##. Besides, I understand that, given a finite vectorspace ##(\Bbb R, V, +)##, any subspace ##U## of ##V## has a complementary...
  9. W

    I Subspace Topology Basics

    Ok, sorry, I am being lazy here. I am tutoring intro topology and doing some refreshers. Were given the subspace topology on [0,1] generated by intervals [a,b) and I need to answer whether under this topology, [0,1] is Hausdorff, Compact or Connected. I think my solutions work , but I am looking...
  10. appletree23

    Help with linear algebra: vectorspace and subspace

    So the reason why I'm struggling with both of the problems is because I find vector spaces and subspaces hard to understand. I have read a lot, but I'm still confussed about these tasks. 1. So for problem 1, I can first tell you what I know about subspaces. I understand that a subspace is a...
  11. Eclair_de_XII

    Cartesian sum of subspace and quotient space isomorphic to whole space

    Let ##n=\dim X## and ##m=\dim Y##. Define a basis for ##X: y_1,...,y_m,z_{m+1},...,z_n##. The first ##m## terms are a basis for ##Y##. The remaining ##n-m## terms are a basis for its complement w.r.t ##X##. Let's call it ##Z##. ##X## is the direct sum of ##Y## and ##Z##; denote it as ##X=Y+Z##...
  12. F

    Finding the dimension of a subspace

    I am stuck on finding the dimension of the subspace. Here's what I have so far. Proof: Let ##W = \lbrace x \in V : [x, y] = 0\rbrace##. We see ##[0, y] = 0##, so ##W## is non empty. Let ##u, v \in W## and ##\alpha, \beta## be scalars. Then ##[\alpha u + \beta v, y] = \alpha [u, y] + \beta [v...
  13. M

    MHB Subsets of $\mathbb{R}^2$ Satisfying S2 and S3 but Not S1: Empty Set

    Hey! :giggle: The three axioms for a subspace are: S1. The set must be not-empty. S2. The sum of two elements of the set must be contained in the set. S3. The scalar product of each element of the set must be again in the set. I have shown that: - $\displaystyle{X_1=\left...
  14. Mayhem

    I Showing that a set of differentiable functions is a subspace of R

    Problem: Show that the set of differentiable real-valued functions ##f## on the interval ##(-4,4)## such that ##f'(-1) = 3f(2)## is a subspace of ##\mathbb{R}^{(-4,4)}## This is my first bouts with rigorous mathematics and my brain is not at all wired for attacking problems like this (yet). I...
  15. forkosh

    A Is the "op" lattice ##\mathscr{L_H}^\perp## also atomistic....?

    Let ##\mathscr{L_H}## be the usual lattice of subspaces of Hilbert space ##\mathscr{H}##, where for ##p,q\in\mathscr{H}## we write ##p\leq q## iff ##p## is a subspace of ##q##. Then, as discussed by, e.g., Beltrametti&Cassinelli https://books.google.com/books?id=yWoq_MRKAgcC&pg=PA98, this...
  16. P

    I Closure in the subspace of linear combinations of vectors

    This is the exact definition and I've summarized it, as I understand it above. Why is it, that for elements in the third subspace, closure will be lost? Wouldn't you still get another vector (when you add two vectors in that subspace), that's still a linear combination of the vectors in the...
    • Phys12
    • Thread
    • closure Combinations Linear Linear combinations Subspace Vectors
    • Replies: 2
    • Forum: Quantum Physics
  17. M

    MHB Show that the subspace U is ϕ^z-invariant

    Hey! 😊 Let $\mathbb{K}$ be a field and let $V$ be a $\mathbb{K}$-vector space. Let $\phi,\psi:V\rightarrow V$ be linear maps, such that $\phi\circ\psi=\psi\circ\phi$. I have shown using induction that if $U\leq_{\phi}V$ (i.e. it $U$ is a subspace and $\phi$-invariant), then...
  18. M

    MHB Eigenspace is a subspace of V - ψ is diagonalizable

    Hey! 😊 Let $\mathbb{K}$ be a field and let $V$ a $\mathbb{K}$-vector space. Let $\phi, \psi:V\rightarrow V$ be linear operators, such that $\phi\circ\psi=\psi\circ\phi$. Show that: For $\lambda \in \text{spec}(\phi)$ it holds that $\text{Eig}(\phi, \lambda )\leq_{\psi}V$. Let...
  19. LCSphysicist

    I Is this a Linear subspace?

    W = {f(t) | f(0) = 2f(1)} The answer say yes, but i don't know how to prove the neutral element.
  20. T

    Subspace of vectors orthogonal to an arbitrary vector.

    The proof that the set is a subspace is easy. What I don't get about this exercise is the dimension of the subspace. Why is the dimension of the subspace ##n-1##? I really don't have a clue on how to go through this.
  21. C

    I Definition of a Degenerate Subspace (QM)

    I was learning about Degenerate Perturbation Theory and I encountered the term 'Degenerate Subspace', I didn't really understand what it meant so I came here to ask - what does it mean? will it matter if i'll say 'Degenerate space' instead of 'Degenerate Subspace'? and subspace of what? (...
    • CGandC
    • Thread
    • Definition Degenerate perturbation theory Perturbation theory Qm Quantum mechanics Subspace
    • Replies: 12
    • Forum: Quantum Physics
  22. M

    MHB Show that it is a subspace

    Hey! :o Let $1\leq m, n\in \mathbb{N}$, let $\phi :\mathbb{R}^n\rightarrow \mathbb{R}^m$ a linear map and let $U\leq_{\mathbb{R}}\mathbb{R}^n$, $W\leq_{\mathbb{R}}\mathbb{R}^m$ be subspaces. I want to show that: $\phi (U)$ is subspace of $\mathbb{R}^m$. $\phi^{-1} (W)$ is subspace of...
  23. M

    Why do we try to find if a subset is a subspace of a vector space?

    I am assuming the set ##V## will have elements like the ones shown below. ## v_{1} = (200, 700, 2) ## ## v_{2} = (250, 800, 3) ## ... 1. What will be the vector space in this situation? 2. Would a subspace mean a subset of V with three or more bathrooms?
  24. N

    MHB Infinite dimentional subspace need not be closed

    Let X=C[O,1] and Y=span($X_{0},X_{1},···$), where $X_{j}={t}^{i}$, so that Y is the set of all polynomials. Y is not closed in X.
  25. S

    Prove that the set T:={x∈Rn:Ax∈S} is a subspace of Rn.

    1. Let's show the three conditions for a subspace are satisfied: Since ##\mathbf{0}\in \mathbb{R}^n##, ##A\times \mathbf{0} = \mathbf{0}\in S##. Suppose ##x_1, x_2\in \mathbb{R}^n##, then ##A(x_1+x_2) = A(x_1)+A(x_2)\in S##. Suppose ##x\in S## and ##\lambda\in \mathbb{R}##, then ##A(\lambda x) =...
  26. J

    I Finding a linear combination to enter a sphere

    Let's say we have n vectors in ℝ3. And say we have defined a subspace inside ℝ3 in the form of a sphere with radius r, and the center of the spheare is at P, where P is a vector in ℝ3. What methods exists to find any linear combination of the n vectors, so that the sum of all of them, lies...
  27. J

    MHB Subspace of three equations

    S is the set of solutions for the set of three equations... x + (1 - a)y-1 + 2z + b2w = 0 ax + y - 3z + (a - a2)|w| = a3 - a x + (a - b)y + z + 2a2w = b I worked out... The first equation is a subset of R4 when a = 1, b is any real. The second equation is a subset of R4 when a = 1 or a = 0...
  28. J

    MHB Three subspace problems

    D is the set and the set contains the solutions to x + (1 - m)y-1 + 2z + n2w = 0 I'm trying to find m, n values which means the set is a subspace of R (four dimensions). === Similarly, trying to find the m, n values that makes the following two expressions two separate subspaces, too. mx +...
  29. V

    Finding a basis for a subspace

    I had assumed that we had to put our values into a matrix so I did [1 2 -1 0; 1 -5 0 -1] and then I would do a=[1; 1] and repeat for b, c, and d. This is incorrect however. I also thought that it could be {(1, 2, -1, 0),(1, -5, 0, -1)} however this was not the answer, and I am unsure of what do...
  30. Kaguro

    I When is a subset a subspace

    Let ##\mathbb{V}## be a vector space and ##\mathbb{W}## be a subset of ##\mathbb{V}##, with the same operations. Claim: If ##\mathbb{W}## is non-empty, closed under addition and scalar multiplication, then ##\mathbb{W}## is a subspace of ##\mathbb{V}##. A set is a vector space if it...
  31. L

    A How to numerically diagonalize a Hamiltonian in a subspace?

    I want to exactly diagonalize the following Hamiltonian for ##10## number of sites and ##5## number of spinless fermions $$H = -t\sum_i^{L-1} \big[c_i^\dagger c_{i+1} - c_i c_{i+1}^\dagger\big] + V\sum_i^{L-1} n_i n_{i+1}$$ here ##L## is total number of sites, creation (##c^\dagger##) and...
    • Luqman Saleem
    • Thread
    • Computational physics Condensed matter physics Diagonalization Hamiltonian Quantum phyics Simulation Subspace
    • Replies: 4
    • Forum: Quantum Physics
  32. GlassBones

    How to show a subspace must be all of a vector space

    Homework Statement Show that the only subspaces of ##V = R^2## are the zero subspace, ##R^2## itself, and the lines through the origin. (Hint: Show that if W is a subspace of ##R^2## that contains two nonzero vectors lying along different lines through the origin, then W must be all of ##R^2##)...
  33. S

    The union of three subspaces of V is a subspace of V

    Homework Statement This is the exact phrasing form Linear Algebra Done Right by Axler: Prove that the union of three subspaces of V is a subspace of V if and only if one of the subspaces contains the other two. [This exercise is surprisingly harder than the previous exercise, possibly because...
  34. Eclair_de_XII

    T is cyclic iff there are finitely many T-invariant subspace

    Homework Statement "Let ##T## be a linear operator on a finite-dimensional vector space ##V## over an infinite field ##F##. Prove that ##T## is cyclic iff there are finitely many ##T##-invariant subspaces. Homework Equations T is a cyclic operator on V if: there exists a ##v\in V## such that...
  35. I

    [Linear Algebra] Linear Transformations, Kernels and Ranges

    Homework Statement Prove whether or not the following linear transformations are, in fact, linear. Find their kernel and range. a) ## T : ℝ → ℝ^2, T(x) = (x,x)## b) ##T : ℝ^3 → ℝ^2, T(x,y,z) = (y-x,z+y)## c) ##T : ℝ^3 → ℝ^3, T(x,y,z) = (x^2, x, z-x) ## d) ## T: C[a,b] → ℝ, T(f) = f(a)## e) ##...
  36. I

    [Linear Algebra] Another question on subspaces

    Homework Statement Let ##V## be the vector space of the sequences which take real values. Prove whether or not the following subsets ##W \in V## are subspaces of ##(V, +, \cdot)## a) ## W = \{(a_n) \in V : \sum_{n=1}^\infty |a_n| < \infty\} ## b) ## W = \{(a_n) \in V : \lim_{n\to \infty} a_n...
  37. G

    2D subspace of a Hilbert space

    Homework Statement Have to read a paper and somewhere along the line it claims that for any distinct ## \ket{\phi_{0}}## and ##\ket{\phi_{1}}## we can choose a basis s.t. ## \ket{\phi_{0}}= \cos\frac{\theta}{2}\ket{0} + \sin\frac{\theta}{2}\ket{1}, \hspace{0.5cm} \ket{\phi_{1}}=...
  38. I

    Determining if a subset W is a subspace of vector space V

    Homework Statement Let V = RR be the vector space of the pointwise functions from R to R. Determine whether or not the following subsets W contained in V are subspaces of V. Homework Equations W = {f ∈ V : f(1) = 1} W = {f ∈ V: f(1) = 0} W = {f ∈ V : ∃f ''(0)} W = {f ∈ V: ∃f ''(x) ∀x ∈ R} The...
  39. bornofflame

    [LinAlg] Find the dimension of the subspace

    Homework Statement Find the dimension of the subspace of all vectors in ##\mathbb{R}^3## whose first and third entries are equal. Homework Equations The Attempt at a Solution So I arrived at two solutions and I'm not entirely sure which is the valid one. #1 Let ##H \text{ be a subspace of...
  40. bornofflame

    [Linear Algebra] Show that H ∩ K is a subspace of V

    Homework Statement From Linear Algebra and Its Applications, 5th Edition, David Lay Chapter 4, Section 1, Question 32 Let H and K be subspaces of a vector space V. The intersection of H and K is the set of v in V that belong to both H and K. Show that H ∩ K is a subspace of V. (See figure.)...
  41. facenian

    I Compact subspace in metric space

    Is there an easy example of a closed and bounded set in a metric space which is not compact. Accoding to the Heine-Borel theorem such an example cannot be found in ##R^n(n\geq 1)## with the usual topology.
  42. K

    I R is disconnected with the subspace topology

    I want to show that ##\mathbb{R}## is disconnected with the subspace topology. For this I considered that ##\mathbb{R} = \lim_{\delta n \longrightarrow 0 } (-\infty, n] \cup [n+\delta n, \infty)## and of course the intersection of these two open sets is empty. What I'm not sure is about the...
  43. facenian

    I Problem about a connected subspace

    Helo, I believe that the folowing exercise from Topology by Munkres is incorrect: "Let A be a proper subset of X, and let B be a proper subsert of Y. If X and Y are conected, show that ##(X\times Y)-(A\times B)## is connected" I think I can prove it wrong however I'm not sure and would like to...
  44. K

    Determine Vector subspace

    Homework Statement Determine the vector subspace generated by ##A = \{x^2 -x, 3 - x^2, 1+x \} \subset P^2(x)## Homework Equations The Attempt at a Solution I tried the usual check of vector addition and scalar multiplication to get the conditions that ##x## and ##y## should satisfy, but...
  45. M

    Show that a set of vectors spans a subspace

    Homework Statement Show that {(1, 2, 3), (3, 4, 5), (4, 5, 6)} does not span R3. Show that it spans the subspace of R3 consisting of all vectors lying in the plane with the equation x - 2y + z = 0. Homework Equations The Attempt at a Solution I made a matrix of: A = [ 1 3 4 ; 2 4 5; 3 5 6]...
  46. SemM

    I How to find admissible functions for a domain?

    Hi, in a text provided by DrDu which I am still reading, it is given that "the momentum operator P is not self-adjoint even if its adjoint ##P^{\dagger}=-\hbar D## has the same formal expression, but it acts on a different space of functions." Regarding the two main operators, X and D, each has...
  47. R

    Is W a subspace of R^3

    Determine whether or not W is a subspace of R^3, where W consists of all vectors (a, b, c) in R^3 such that (a) a = 3b; (b) a<=b<=c; (c) ab = 0. a) Because vectors (a,b,c) can assume any value in W, W is a subset of R^3. Also, the zero vector belongs to W and W is closed under vector addition...
  48. C

    Orthogonal projection over an orthogonal subspace

    Homework Statement Being F = (1,1,-1), the orthogonal projection of (2,4,1) over the orthogonal subspace of F is: a) (1,2,3) b) (1/3, 7/3, 8/3) c) (1/3, 2/3, 8/3) d) (0,0,0) e) (1,1,1) The correct answer is B Homework Equations The Attempt at a Solution Using the orthogonal projection...
  49. N

    Linear Algebra subspace help

    Homework Statement I have an assignment for my linear algebra class, that I simply cannot figure out. Its going to be hard to follow the template of the forum, as its a rather simply problem. It is as follows: Given the following subspace (F = reals and complex) and the "linear image"...
  50. M

    MHB Intersection of all subspace of V is the empty set

    Hey! :o Let $V$ be a $\mathbb{R}$-subspace with basis $B=\{v_1 ,v_2, \ldots , v_n\}$ and $\overline{v}\in V$, $\overline{v}\neq 0$. I have shown that if we exchange $\overline{v}$ with an element $v_i\in B$ we get again a basis. How can we show, using this fact, that the intersection of all...
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