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. 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...
  2. S

    Find the basis of a vector subspace of R^2,2

    Homework Statement i know how to find the basis of a subspace of R2 or R3 but I can't figure out how to find the basis of a subspace of something like R2,2. I even have an example in my book which i managed to follow nearly till the end but not quite... Given matrix: A= 6 -9 4 -4 show that...
  3. Y

    MHB What Is Required to Prove a Subset is a Vector Space?

    Hello all, I have a theoretical question regarding subspaces. If V is a subset of a vector space, and we wish to show that V is a vector space itself, we need to show 3 things. Some references say we need to show: a) V is not empty b) V is closed under + c) V is closed under scalar...
  4. B

    I Understanding Hilbert Subspace for Two-Particle Entangled Systems

    I read that if we construct an observable on a two-particle entangled system like the "center of mass" observable, this observable does not pick out a single state of the two-particle system. It only picks out a subspace of the full Hilbert space of all possible states--the subspace that...
  5. PsychonautQQ

    Is Y Union A Connected When X Minus Y Splits into Disjoint Subsets A and B?

    Homework Statement Let Y be a subspace of X and let both X and Y be connected. If X-Y=AUB where the intersection of A and B is empty, show that YUA is connected. Homework EquationsThe Attempt at a Solution Say YUA = CUD where C and D are disjoint. Let C_y be the intersection of Y with C and...
  6. tze liu

    Is it possible that a subspace is not a vector space

    <Mentor's note: moved from general mathematics to homework. Thus no template.> Prove subspace is only a subset of vector space but not a vector space itself. Even a subspace follows closed under addition or closed under multiplication,however it is not necessary to follow other 8 axioms in...
  7. Adgorn

    I Proving a lemma on decomposition of V to T-cyclic subspace

    I am reading Schaum's outlines linear algebra, and have reached an explanation of the following lemma: Let ##T:V→V## be a linear operator whose minimal polynomial is ##f(t)^n## where ##f(t)## is a monic irreducible polynomial. Then V is the direct sum ##V=Z(v_1,T)⊕...⊕Z(v_r,T)## of T-cyclic...
  8. A

    How can you tell if somethign is a subspace with abstract info?

    Homework Statement http://prntscr.com/ej0akz Homework EquationsThe Attempt at a Solution I know there are three problems in one here, but they are all of the same nature. I don't understand how this is enough information to find out if they are subspaces. It's all really abstract to me. I know...
  9. R

    I Exploring 2D Vector Subspaces: A Question of Multidimensional Components

    If each component of a 2D vector have multiple variables/dimensions, does that make two subspaces? For example the vector (1 - b/a, 1 - a/b) Can I convert both components surfaces into two 3D surfaces in the same space and find their intersection?
  10. BubblesAreUs

    Determine whether the subset is a vector subspace

    Homework Statement Recall that F is the vector space of functions from ℝ to ℝ, with the usual operations of addition and scalar multiplication of functions. For each of the following subsets of F, write down two functions that belong to the subset, and determine whether or not the subset is a...
  11. F

    I Basis of a Subspace of a Vector Space

    Hello Forum and happy new year, Aside from a rigorous definitions, a linear vector space contains an infinity of elements called vectors that must obey certain rules. Based on the dimension ##N## (finite or infinite) of the vector space, we can always find a set of ##n=N## linearly independent...
  12. C

    MHB Showing that P_3 is a subspace to P_n

    Dear everyone, I need to show that the $${P}_{3}$$ is a Subspace to $${p}_{n}$$. how to start the proof?
  13. M

    MHB Is there a more efficient way to determine the basis of a subspace?

    Hey! :o We are given the vectors $\vec{a}=\begin{pmatrix}4\\ 1 \\ 0\end{pmatrix}, \vec{b}=\begin{pmatrix}2\\ 0 \\ 1\end{pmatrix}, \vec{c}=\begin{pmatrix}0\\ -2 \\ 4\end{pmatrix}$. I have shown by calculating the deteminant $|D|=0$ that these three vectors are linearly dependent. I want to...
  14. TheSodesa

    A real parameter guaranteeing subspace invariance

    Homework Statement Let ##A## and ##B## be square matrices, such that ##AB = \alpha BA##. Investigate, with which value of ##\alpha \in \mathbb{R}## the subspace ##N(B)## is ##A##-invariant. Homework Equations If ##S## is a subspace and ##A \in \mathbb{C}^{n \times n}##, we define multiplying...
  15. Mr Davis 97

    I Orthogonal basis to find projection onto a subspace

    I know that to find the projection of an element in R^n on a subspace W, we need to have an orthogonal basis in W, and then applying the formula formula for projections. However, I don;t understand why we must have an orthogonal basis in W in order to calculate the projection of another vector...
  16. Mr Davis 97

    Dimension and basis for a subspace

    Homework Statement ##\mathbb{H} = \{(a,b,c) : a - 3b + c = 0,~b - 2c = 0,~2b - c = 0 \}## Homework EquationsThe Attempt at a Solution This definition of a subspace gives us the vector ##(3b - c,~2c,~2b) = b(3,0,2) + c(-1,2,0)##. This seems to suggest that a basis is {(3, 0, 2), (-1, 2 0)}, and...
  17. Mr Davis 97

    Upper trianglar matrix is a subspace of mxn matrices

    Homework Statement Prove that the upper triangular matrices form a subspace of ##\mathbb{M}_{m \times n}## over a field ##\mathbb{F}## Homework EquationsThe Attempt at a Solution We can prove this entrywise. 1) Obviously the zero matrix is an upper triangular matrix, because it satisfies the...
  18. Mr Davis 97

    Proving that a subset is a subspace

    Homework Statement Determine whether ##W = \{(a_1, a_2, a_3) \in \mathbb{R}^3 : a_1 = 3a_3,~ a_3 = -a_2 \}## is a subspace of ##\mathbb{R}^3##. Homework EquationsThe Attempt at a Solution To show that a subset of vector space is a subspace we need to show three things: 1) That the zero vector...
  19. Mr Davis 97

    I Nonempty Subspace: Proving 0u = 0

    I have a simple question. Say we have some subspace that is nonempty and closed under scalar multiplication and vector addition. How could we deduce that ##0 \vec{u} = \vec{0}##?
  20. Saph

    Prove :Union of Three subspaces is a subspace if ....

    Homework Statement Prove the the union of three subspaces is a subspace if one of the subspaces contains the others Homework Equations A subset W of a vector space V is called a subspace if : 1) ##0 \in W ##. 2) if ##U_1## and ##U_2## are in ##W##, then ##U_1 + U_2 \in W##, 3) if ##\alpha ##...
  21. M

    I Understanding Subspaces: Definition and Examples | PF Study Guide

    Hi PF! I want to make sure I understand the notion of a subspace. Our professor gave an example of one: the set of degree n polynomials is a subspace of continuous functions. This is because a) a polynomial is intrinsically a subset of continuous functions and b) summing any polynomials yields...
  22. S

    Does V Qualify as a Subspace of ℝ3?

    Homework Statement Let X=ℝ3 and let V={(a,b,c) such that a2+b2=c2}. Is V a subspace of X? If so, what dimensions? Homework Equations A vector space V exists over a field F if V is an abelian group under addition, and if for each a ∈ F and v ∈ V, there is an element av ∈ V such that all of...
  23. ibkev

    Linear Algebra showing a subspace

    I'm having trouble getting started on this one and I'd really appreciate some hints. This question comes from Macdonald's Linear and Geometric Algebra book that I'm using for self study, problem 2.2.4. Homework Statement Let U1 and U2 be subspaces of a vector space V. Let U be the set of all...
  24. Z

    MHB A subset of R^(2) that is a nontrivial subspace of R^(2)

    Hey, I need help coming up an example of a subset of $\Bbb R^2$ that is a nontrivial subspace of $\Bbb R^2$. Thank you!
  25. Q

    I Modelling Decoherence as a Transition Out of Subspace

    Modelling the onset of decoherence in a subspace as a transition from this subspace http://www.ba.infn.it/~pascazio/publications/sudarshan_seven_quests.pdf (Section 10 is relevant) I am currently reading papers discussing the Zeno Effect. The linked paper discusses modelling a transition out of...
  26. Q

    I Keeping a subspace decoherence free, with the Zeno Effect

    I am currently reading papers discussing the Zeno Effect, which discuss how measuring a system at high frequencies can almost freeze the state of a system, or keep the system in a specific subspace of states. This can be easily seen using the projection postulate. Often the topic of decoherence...
  27. Prof. 27

    Showing that Something is a Subspace of R^3

    Homework Statement The question asks to show whether the following are sub-spaces of R^3. Here is the first problem. I want to make sure I'm on the right track. Problem: Show that W = {(x,y,z) : x,y,z ∈ ℝ; x = y + z} is a subspace of R^3. Homework Equations None The Attempt at a Solution...
  28. G

    MHB Subspace spanned by subsets of polynomials

    In the linear space of all real polynomials $p(t)$, describe the subspace spanned by each of the following subsets of polynomials and determine the dimension of this subspace. (a) \left\{1,t^2,t^4\right\}, (b) \left\{t,t^3,t^4\right\}, (c) \left\{t,t^2\right\}, (d) $\left\{1+t, (1+t)^2\right\}$...
  29. G

    MHB Dimension & Subspace of $\mathbb{R}^3$

    Check whether the following are subspaces of $\mathbb{R}^3$ and if they're find their dimension. (a) x = 0, (b) x+y = 0, (c) x+y+z = 0, (d) x = y, (e) x = y= z, and (f) x = y or x = z. (a) Let $S = \left\{(x, y, z) \in \mathbb{R}^3:x = 0 \right\}$. I want to check whether $S$ is subspace of...
  30. G

    Linear algebra: Prove that the set is a subspace

    Homework Statement Let U is the set of all commuting matrices with matrix A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 1 \\ 3 & 0 & 4 \\ \end{bmatrix}. Prove that U is the subspace of \mathbb{M_{3\times 3}} (space of matrices 3\times 3). Check if it contains span\{I,A,A^2,...\}. Find the...
  31. M

    How to determine the smallest subspace?

    Two examples are: Given two subspaces ##U## and ##W## in a vector space ##V##, the smallest subspace in ##V## containing those two subspaces mentioned in the beginning is the sum between them, ##U+W##. The smallest subspace containing vectors in the list ##(u_1,u_2,...,u_n)## is...
  32. rayne1

    MHB Show that V is a subspace of P2

    $$V = \{({x}^{2}-1)p(x) | p(x) \in {P}_{2}\}$$ show that V is a subspace of ${P}_{2}$I tried: $({x}^{2}-1)(0) = 0$ so 0 is in ${P}_{2}$ (axiom 1 is satisfied). If p(x) and q(x) are in ${P}_{2}$, then $({x}^{2}-1)p(x) + ({x}^{2}-1)q(x) = ({x}^{2}-1)(p(x)+q(x))$ and since $p(x)+q(x) \in...
  33. Q

    Vector Subspace: Show S is a Subspace, Determine Basis & Find Dimension

    Homework Statement Can you help please? I have this problem: Let S be the set of all vectors parallel to the hyper-plane 4x +2y+z + 3r =0 in R^4 . (a) Show that S is a subspace, (b) Determine a basis for S , (c) Find its dimension Homework EquationsThe Attempt at a Solution S= { u=(x, y,z,r) |...
  34. M

    Is this matrix a subspace?

    Homework Statement \begin{pmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} Is this set a subspace of ℝ3 Homework Equations The set must be closed under addition. The set must be closed under multiplication. The set must contain the zero vector. The Attempt at a...
  35. H

    Connection between subspace, span, and basis?

    I'm stuck on a relation issue if there is a direct relation at all. If I were to verify that a subset is a subspace of a vector space V, would it then be correct to check that subset for linear independence to verify that the subset spans the subspace? I'm not sure if I'm following the...
  36. Andrew Pierce

    Determining subspaces for all functions in a Vector space

    Homework Statement First, I'd like to say that this question is from an Introductory Linear Algebra course so my knowledge of vector space and subspace is limited. Now onto the question. Q: Which of the following are subspaces of F(-∞,∞)? (a) All functions f in F(-∞,∞) for which f(0) = 0...
  37. kostoglotov

    What subspace of 3x3 matrices is spanned by rank 1 matrices

    So that's the question in the text. I having some issues I think with actually just comprehending what the question is asking me for. The texts answer is: all 3x3 matrices. My answer and reasoning is: the basis of the subspace of all rank 1 matrices is made up of the basis elements...
  38. L

    Proving that a subspace of C is a field

    Homework Statement Show that the only m for which the subspace of C given by {z ∈ C: Im(z) = m Re(z)} is a field is m=0. Homework Equations Field axioms The Attempt at a Solution I tried to prove one direction : - If z is in the subspace, Re z>0 and m≠0 then Arg z<Arg z^2, so z^2 is not in...
  39. Hugo S

    Determine whether a subset is a subspace

    Determine whether the following subset is a linear subspace of ##F^3##. ## X = \left\{ (x_1, x_2, x_3) \in \mathbb{F^3}:x_1 x_2 x_3 = 0 \right\} ## I know that I can simply provide a counterexample and show that the subset X above is not closed under addition -- namely, I can construct two...
  40. S

    Smallest subspace of a vector space

    I may have a bad day, or not enough coffee yet. So, "If A is a nonempty subset of a vector space V, then the set L(A) of all linear combinations of the vectors in A is a subspace, and it is the smallest subspace of V which includes the set A. If A is infinite, we obviously can't use a single...
  41. Ahmad Kishki

    Not (null T) is a subspace of V?

    If T is a linear operator L(V,W) then can we say that all the vectors (in the vector space V) that T does not map to the zero vector (in the vector space W) form a subspace call it X? If a collection of vectors forms a subspace then they must satisfy closure under vector addition and scalar...
  42. S

    Double Orthogonal Closed Subspace Inner Product => Hilbert

    Let X be an Inner Product Space. If for every closed subspace M, M^{\perp \perp} = M, then X is a Hilbert Space (It's complete). Hint: Use the following map: T : X \longrightarrow \overset{\sim}{X}: T(y)=(x,y)=f(x) where (x,y) is the inner product of X. Relevant equations: S^{\perp} is always...
  43. M

    Linear algebra problem related to vector subspace

    Homework Statement X ={(x1,x2,x2 −x1,3x2):x1,x2 ∈R} f(x1,x2,x2 −x1,3x2)=(x1,x1,0,3x1) 1. Find a basis for X. 2. Find dim X. 3. Find ker f and I am f 4. Find bases for ker f and I am f 5. Is f a bijection? Why? 6. Find a diagonal matrix for f.Homework EquationsThe Attempt at a Solution 1. Put...
  44. M

    Vector subspace and linear transformation

    X ={(x1,x2,x2 −x1,3x2):x1,x2 ∈R} f(x1,x2,x2 −x1,3x2)=(x1,x1,0,3x1) 1. Find a basis for X. 2. Find dim X. 3. Find ker f and I am f 4. Find bases for ker f and I am f 5. Is f a bijection? Why? 6. Find a diagonal matrix for f. My attempt: 1. (1, 1, 0, 3) and (1, 2, 1, 6) 2. Dim X = 2 3. Ker f = 0...
  45. blue_leaf77

    Subspace of a 4x4 matrix space

    Homework Statement This question is taken from Linear Algebra Done Wrong by Treil. Question 7.5 of chapter 1 says this: What is the smallest subspace of the space of 4 4 matrices which contains all upper triangular matrices (aj,k = 0 for all j > k), and all symmetric matrices (A = AT )? What is...
  46. Dethrone

    MHB Is $U$ a Subspace of $F[0,1]$?

    Is the following a subspace of $F[0,1]$? $U={}\left\{f|f(0)=f(1)\right\}$ First, it contains the 0 vector if you consider $f(x)=0$, which is 0 for all $x$. Now I'm not sure how to prove that it is closure under addition. Here's what I have so far: If $f_1, f_2 \in U$, then...
  47. 7

    Determine if all vectors of form (a,0,0) are subspace of R3

    I have the feeling that it is, but I am not really sure how to start the proof. I know I have to prove both closure axioms; u,v ∈ W, u+v ∈ W and k∈ℝ and u∈W then ku ∈ W. Do I just pick a vector arbitrarily say a vector v = (x,y,z) and go from there?
  48. N

    MHB Proving Subspace: Basic Proof for $V(A)$

    Hi everyone, would really appreciate if someone could help me with the attached question (its the one in the red box). My start: Assume $A$ is a subspace. We need to show that $V(A):= \{ \{x_n\}_n \in V_0 : \lim_{{n}\to{\infty}}x_n \in A \} $ By definition, a subspace is closed under...
  49. M

    Subspace / basis problem

    Homework Statement Let S, a subspace of ℝ3 be the set of vectors orthogonal to vector (1,2,3) a)describe Set S b) find a basis for Set S 2. Relevant Equations That a basis has to be linearly independent and span R^3The Attempt at a Solution [/B] I would do this: I know that vector (1,2,3) is...
  50. H

    Finding the basis of a subspace

    Homework Statement How do I find a basis for: the subspace of R^3 consisting of all vectors x such that x ⋅ (1,2,3) = 0. Homework Equations I believe this is performed through setting x = x,y,z, setting each parameter sequentially equal to 1 while the others are set to o, putting into a matrix...
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