What is Subspaces: Definition and 333 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. cianfa72

    I Homeomorphism linear subspace

    Hi, consider the Euclidean space ##\mathbb R^8## and the projection map ##\pi## over the first 4 coordinates, i.e. ##\pi : \mathbb R^8 \rightarrow \mathbb R^4##. I would show that the restriction of ##\pi## to the linear subspace ##A## (endowed with the subspace topology from ##\mathbb R^8##)...
  2. J

    Sets of Vectors as Subspaces

    I don't understand the solution: that for (1, ..., 1) the additive inverse is (-1, ..., -1), so the condition is not satisfied (and it is not a subspace). Which condition is not met? Thank you.
  3. docnet

    Prove that the intersection of subspaces is compact and closed

    Given that one of the ##S_i## (let's name it ##S_{compact}##), is compact. Assume there is an open cover ##\mathcal V## of ##S_{compact}##. By definition of a compact subspace, there is a subcover ##\mathcal U## with ##n<\infty## open sets. Notice that ##\forall x\in (\bigcap_i S_i)##, ##x\in...
  4. JD_PM

    Dimension statement about (finite-dimensional) subspaces

    My intuition tells me this is a true statements so let's try to prove it. The dimension is defined as the number of elements of a basis. Hence, we can work in terms of basis to prove the statement. Given that ##U_3## appears on both sides of the inequality, let's get a basis for it. How...
  5. JD_PM

    Proving a statement about inclusion of subspaces

    Summary:: I want to understand the following theorem and its proof (which can be found in MSE, link below): Let ##V## be a ##n##-dimensional vector space, let ##U_i \subseteq V## be subspaces of ##V## for ##i = 1,2,\dots,r## where $$U_1 \subseteq U_2 \subseteq \dots \subseteq U_r$$ If ##r>n+1##...
  6. JD_PM

    Given subspaces ##U \& W##, show they are equal | Linear Algebra

    Show that ##U = span \{ (1, 2, 3), (-1, 2, 9)\}## and ##W = \{ (x, y, z) \in \Bbb R^3 | z-3y +3x = 0\}## are equal. I have the following strategy in mind: determine the dimension of subspaces ##U## and ##W## separately and then make use of the fact ##dim U = dim W \iff U=W##. For ##U## I would...
  7. JD_PM

    Span(S) is the intersection of all subspaces of V containing S

    Homework Statement:: I want to understand the proof for the following theorem: span(S) is the intersection of all subspaces of V containing S. Relevant Equations:: N/A I know that if ##W## is any subspace of ##V## containing ##S## then ##\text{span}(S) \subseteq W##. I have read (Page 157: #...
  8. V

    Dimension of orthogonal subspaces sum

    ##| V_1 \rangle \in \mathbb{V}^{n_1}_1## and there is an orthonormal basis in ##\mathbb{V}^{n_1}_1##: ##|u_1\rangle, |u_2\rangle ... |u_{n_1}\rangle## ##| V_2 \rangle \in \mathbb{V}^{n_2}_2## and there is an orthonormal basis in ##\mathbb{V}^{n_2}_2##: ##|w_1\rangle, |w_2\rangle ...
  9. M

    MHB Statements about subspaces

    Hey! :giggle: Let $V$ be a $\mathbb{R}$-vector space, let $x,y\in V$ and let $U,W\leq_{\mathbb{R}}V$ be subspaces of $V$. Show that : (a) If $(x+U)\cap (y+W)\neq \emptyset$ and $z\in (x+U)\cap (y+W)$ then $(x+U)\cap (y+W)=z+(U\cap W)$. (b) The following statements are equivalent: (i) $U=W$ and...
  10. G

    Subspace Help: Properties & Verifying Examples

    Summary:: Properties of subspaces and verifying examples Hi, My textbook gives some examples relating to subspaces but I am having trouble intuiting them. Could someone please help me understand the five points they are attempting to convey here (see screenshot).
  11. J

    B Are subspaces of Hilbert space real?

    When orthogonal states of a quantum system is projected into subspaces A and B are A and B real spaces?
  12. nomadreid

    I Quantum logic based on closed Hilbert space subspaces

    One proposal that I have read (but cannot re-find the source, sorry) was to identify a truth value for a proposition (event) with the collection of closed subspaces in which the event had a probability of 1. But as I understand it, a Hilbert space is a framework which, unless trivial, keeps...
  13. N

    I 2 and 3 dimensional invariant subspaces of R4

    I am looking at the representation of D4 in ℝ4 consisting of the eight 4 x 4 matrices acting on the 4 vertices of the square a ≡ 1, b ≡ 2, c ≡ 3 and d ≡ 4. I have proven that the 1-dimensional subspace of D4 in ℝ2 has no proper invariant subspaces and therefore is reducible. I did this in 2...
  14. M

    MHB Decide if the sets are subspaces or affine subspaces

    Hey! :o We have the subsets \begin{equation*}V:=\left \{\begin{pmatrix}x_1 \\ x_2 \\ x_3\end{pmatrix}\mid x_1=0\right \}, \ \ \ W:=\left \{\begin{pmatrix}x_1 \\ x_2 \\ x_3\end{pmatrix}\mid x_2=2\right \}, \ \ \ S:=\left \{\lambda \begin{pmatrix}1 \\ 0 \\ -1\end{pmatrix}\mid \lambda \in...
  15. M

    MHB Subset that satisfies all but one axioms of subspaces

    Hey! :o I want to find subsets $S$ of $\mathbb{R}^2$ such that $S$ satisfies all but one axioms of subspaces. A subset that doesn't satisfy the first axiom: We have to find a subset that doesn't contain the zero vector. Is this for example $\left \{\begin{pmatrix}x \\ y\end{pmatrix} ...
  16. M

    MHB Are these sets subspaces?

    Hey! :o We have the following subsets: \begin{align*}&U_1:=\left \{\begin{pmatrix}x \\ y\end{pmatrix} \mid x^2+y^2\leq 4\right \} \subseteq \mathbb{R}^2\\ &U_2:=\left \{\begin{pmatrix}2a \\ -a\end{pmatrix} \mid a\in \mathbb{R}\right \} \subseteq \mathbb{R}^2 \\ &U_3:=\left \{\begin{pmatrix}x...
  17. S

    I Showing direct sum of subspaces equals vector space

    If one shows that ##U\cap V=\{\textbf{0}\}##, which is easily shown, would that also imply ##\mathbf{R}^3=U \bigoplus V##? Or does one need to show that ##\mathbf{R}^3=U+V##? If yes, how? By defining say ##x_1'=x_1+t,x_2'=x_2+t,x_3'=x_3+2t## and hence any ##\textbf{x}=(x_1',x_2',x_3') \in...
  18. Calculuser

    I Confusion about the Direct Sum of Subspaces

    In "Sheldon Axler's Linear Algebra Done Right, 3rd edition", on page 21 "internal direct sum", or direct sum as the author uses, is defined as such: Following that there is a statement, titled "Condition for a direct sum" on page 23, that specifies the condition for a sum of subspaces to be...
  19. Math Amateur

    I Bases for Tangent Spaces and Subspaces - McInerney Theorem 3.3.14

    I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.3: Geometric Sets and Subspaces of ##T_p ( \mathbb{R}^n )## ... ... I need help with...
  20. Joppy

    MHB Nested linear subspaces

    Without providing further context, is it possible to give meaning to the construction of "nested linear subspaces"? e.g. in $\mathbb{R}^n$ the tangent space at a point $T \mathbb{R}^n = \mathbb{R}^n_x = V_0 \subset V_1 \subset ... \subset V_k $ is the same as saying that the tangent space can...
  21. Math Amateur

    I Tangent Spaces and Subspaces .... McInerney Theorem 3.3.13 ....

    I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.3: Geometric Sets and Subspaces of ##T_p ( \mathbb{R}^n )## ... ... I need help with an aspect...
  22. GlassBones

    Formulation of a proof of subspaces

    Homework Statement Let W be a subspace of a vector space V, let y be in V and define the set y + W = \{x \in V | x = y +w, \text{for some } w \in W\} Show that y + W is a subspace of V iff y \in W. Homework Equations The Attempt at a Solution Let W be a subspace of a vector space V, let y...
  23. Y

    MHB Proving Vector Subspaces of R^3

    Dear all, I am trying to find if these two sets are vector subspaces of R^3. \[V=\left \{ (x,y,z)\in R^{3}|(x-y)^{2}+z^{2}=0 \right \}\] \[W=\left \{ (x,y,z)\in R^{3}|(x+1)^{2}=x^{2}+1 \right \}\] In both cases the zero vector is in the set, therefore I just need to prove closure to addition...
  24. 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...
  25. gibberingmouther

    I Subspaces in Axler's Famous Textbook

    Just started working through "Linear Algebra Done Right". There is something I don't understand. Given b ∈ F, then {(x1,x2,x3,x4) ∈ F4 : x3 = 5x4 + b} is a subspace of F4 *if and only if* b=0 I just flat out don't understand why b has to be 0 or even what is the point of bringing this up...
  26. Pushoam

    Dimensionality of the sum of subspaces

    Homework Statement Suppose that ## \mathbb {V}_1^{n_1} ## and ## \mathbb {V}_2^{n_2} ## are two subspaces such that any element of ## \mathbb {V}_1^{n_1} ## is orthogonal to any element of ## \mathbb {V}_2^{n_2} ## . Show that dimensionality of ## \mathbb {V}_1^{n_1} + \mathbb {V}_2^{n_2}...
  27. I

    [Linear Algebra] Sum & Direct Sum of Subspaces

    ⇒Homework Statement [/B] Calculate ##S + T## and determine if the sum is direct for the following subspaces of ##\mathbf R^3## a) ## S = \{(x,y,z) \in \mathbf R^3 : x =z\}## ## T = \{(x,y,z) \in R^3 : z = 0\}## b) ## S = \{(x,y,z) \in \mathbf R^3 : x = y\}## ## T = \{(x,y,z) \in \mathbf R^3 ...
  28. 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...
  29. I

    [Linear Alg] Determining which sets are subspaces of R[x]

    Homework Statement [/B] Which of the following sets are subspaces of ##R[x]?## ##W_1 = {f \in \mathbf R[x] : f(0) = 0}## ##W_2 = {f \in \mathbf R[x] : 2f(0) = f(1)}## ##W_3 = {f \in \mathbf R[x] : f(t) = f(1-t) \forall t \in \mathbf R}## ##W_4 = {f \in \mathbf R[x] : f = \sum_{i=0}^n...
  30. 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...
  31. bornofflame

    [LinAlg] Set Notation, Subspaces, Bases, Ranges, & Kernels

    Homework Statement I don't want to clog up the forums with a few "small" problems so I am lumping them together here. 2. Let ##T:P^1 → R \text { be given by } T(p(x)) = \int^b_a p(x)dx##. Describe Ker(T) using set notation. 3. Let ##H = \left\{f ∈ C[a, b] | f'(x) ≥ 0 ~\text for...
  32. K

    I Vector spaces and subspaces

    Does it make sense to say that a set together with a field generates a vector space? I came across this question after starting the thread https://www.physicsforums.com/threads/determine-vector-subspace.941424/ To be more specific, suppose we have a set consisting of two elements ##A = \{x^2, x...
  33. U

    Proving S is a Subset of T in R³

    Homework Statement Show that S ⊆ T, where S and T are both subsets of R^3. Homework Equations S = {(1, 2, 1), (1, 1, 2)}, T ={(x,y,3x−y): x,y∈R} The Attempt at a Solution I considered finding if S is a spanning set for T but I'm aware that this is perhaps not relevant. If I find {α(1, 2, 1)...
  34. R

    Understanding Subspaces: Criteria and Examples (R3, Integers, R2)

    Homework Statement Homework EquationsThe Attempt at a Solution So for a subspace 2 criteria have to be met: its closed under addition, and scalar multiplication. Now I have the question "Which of the following subsets of R3 are subspaces? The set of all vectors of the form (a,b,c) where a,b...
  35. E

    I Developing Intuition Behind Affine Subspaces: A Sample Problem

    Hey, I am struggling with developing an intuition behind 'Affine Subspaces'. So far I have read the theories concerning Affine Subspaces delivered by the course book and visited several websites, however none have made it 100% clear. I feel like I have some sort of intuition for it, but I fail...
  36. Austin Chang

    I Prove that V is the internal direct sum of two subspaces

    Let V be a vector space. If U 1 and U2 are subspaces of V s.t. U1+U2 = V and U1 and U1∩U2 = {0V}, then we say that V is the internal direct sum of U1 and U2. In this case we write V = U1⊕U2. Show that V is internal direct sum of U1 and U2if and only if every vector in V may be written uniquely...
  37. S

    Verifying Subspace of P3: Closure of Addition & Scalar Multiplication

    Homework Statement Determine if the following is a subspace of ##P_3##. All polynomials ##a_0+a_1x+a_2x^2+a_3x^3## for which ##a_0+a_1+a_2+a_3=0## Homework Equations use closure of addition and scalar multiplication The Attempt at a Solution Let ##P=a_0+a_1x+a_2x^2+a_3x^3## and...
  38. B

    B Can Everett Branches Break Quantum Laws?

    Are Everett branches (or relative states) the eigenstates or the Hilbert subspaces (or others?)? Once in a branch (or world), what law of QM would be broken if you can cut off the branch you are sitting on and revert back to the global state vector (isn't the quantum eraser, etc. about...
  39. 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...
  40. Mr Davis 97

    Describing the intersection of two subspaces

    Homework Statement ##W_1 = \{(a_1, a_2, a_3) \in \mathbb{R}^3 : a_1 = 3a_3,~ a_3 = -a_2 \}## ##W_2 = \{(a_1, a_2, a_3) \in \mathbb{R}^3 : 2a_1 - 7a_2 + a_3 = 0 \}## Given that these are two subspaces of ##\mathbb{R}^3##, describe the intersection of the two, i.e. ##W_1 \cap W_2## and show that...
  41. 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 ##...
  42. Kernul

    Intersection of two subspaces

    Homework Statement I have this exercise that tells me to determine a base and the dimension of the subspaces of ##\mathbb {R}^4##, ##U \cap Ker(f)## and ##U + Ker(f)##, knowing that: ##U = <\begin{pmatrix} -10 \\ 11 \\ 2 \\ 9 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ 1 \\ 3 \end{pmatrix}...
  43. G

    MHB Linear Subspaces: Properties and Examples

    For the brief explanation: $\mathcal{P}$ contains $0$ by choice $p(x) = 0$ and polynomial plus a polynomial is a polynomial, and a scalar times a polynomial is a polynomial. So $\mathcal{P}$ is a non-empty subset of $\mathcal{C}^{\infty}$ that's closed under addition and scalar multiplication...
  44. F

    I Can subspaces be used to determine probabilities in quantum mechanics?

    Suppose we have an observable with a certain number of eigenstates. We would normalize all these possibilities to 1 in order to give each eigenstate an appropriate probability of being measured. Can we then only consider the data of many measurements for only a subset of those eigenstates and...
  45. 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...
  46. G

    Modular arithmetic on vector spaces

    Homework Statement Let U is the set of all polynomials u on field \mathbb F such that u(3)=u(-2)=0. Check if U is the subspace of the set of all polynomials P(x) on \mathbb F and if it is, determine the set W such that P(x)=U\oplus W. Homework Equations -Polynomial vector spaces -Subspaces...
  47. G

    Linear algebra: Prove the statement

    Homework Statement Prove that \dim L(\mathbb F)+\dim Ker L=\dim(\mathbb F+Ker L) for every subspace \mathbb{F} and every linear transformation L of a vector space V of a finite dimension. Homework Equations -Fundamental subspaces -Vector spaces The Attempt at a Solution Theorem: [/B]If...
  48. Math Amateur

    Geometric Sets and Tangent Subspaces - McInnerney, Example 3

    I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... I am currently focussed on Chapter 3: Advanced Calculus ... and in particular I am studying Section 3.3 Geometric Sets and Subspaces of T_p ( \mathbb{R}^n ) ... I need help with a...
  49. S

    Isomorphism to subspaces of different dimensions

    Homework Statement Given the linear transformations f : R 3 → R 2 , f(x, y, z) = (2x − y, 2y + z), g : R 2 → R 3 , g(u, v) = (u, u + v, u − v), find the matrix associated to f◦g and g◦f with respect to the standard basis. Find rank(f ◦g) and rank(g ◦ f), is one of the two compositions an...
  50. Math Amateur

    MHB Sum of Two Subspaces and lub - Roman, Chapter 1, page 39

    I am reading Steven Roman's book, Advanced Linear Algebra and am currently focussed on Chapter 1: Vector Spaces ... ... In discussing the sum of a set of subspaces Roman writes (page 39) ...In the above text, Roman writes: " ... ... It is not hard to show that the sum of any collection of...
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