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. caffeinemachine

    MHB Bilinear Form Non-Degenerate on a Subspace.

    I am trying to prove the following standard result:Let $V$ be a finite dimensional vector space over a field $F$ and $f:V\times V\to F$ be a symmetric bilinear form on $V$. Let $W$ be a subspace of $V$ such that $f$ is non-degenerate on $W$. Then $$V=W\oplus W^\perp$$(Here $W^\perp=\{v\in...
  2. A

    Finding a basis for a particular subspace with Dot Product restrictions

    Find the basis of the subspace of R4 that consists of all vectors perpendicular to both [1, -2, 0, 3] and [0,2,1,3]. My teacher applies dot product: Let [w,x,y,z] be the vectors in the subspace. Then, w-2x+3z=0 and 2x+y+3z=0 So, she solves the system and get the following: Subspace= {...
  3. M

    Is Every Set in ℝ3 a Subspace?

    Homework Statement A. {(x,y,z)ι x<y<z } B.{(x,y,z)ι -4x+2y=0, -5x-7z=0 } C.{(x,y,z)ι -9x-3y+8z=7} D.{-7x-8y,9x+6y,3x-6y ι x,y arbitrary numbers} E.{(x,y,z)ι x+y+z=0 } F.{(x,x+4,x-2 } Homework Equations So it is a subspace therefore I need the additive axiom: u+v = v+u, the one where u...
  4. Aristotle

    Need help with proof of Vector Space (Ten Axioms)

    Let S={x ∈ R; -π/2 < x < π/2 } and let V be the subset of R2 given by V=S^2={(x,y); -π/2 < x < π/2}, with vector addition ( (+) ). For each (for every) u ∈ V, For each (for every) v ∈ V with u=(x1 , y1) and v=(x2,y2) u+v = (arctan (tan(x1)+tan(x2)), arctan (tan(y1)+tan(y2)) )Note: The...
  5. JohnPrior3

    Subspace Orthogonality in Ax=b

    Let A be the matrix [2 0 1 0 1 -1 4 3 3 -1 5 3] Let b= [b1 b2 b3] transpose What equation must be satisfied by the components of b in order to guarantee that there will exists a vector x= [x1 x2 x3 x4] transpose satisfying the equation Ax=b. Justify your answer. I know C(A) is the orthogonal...
  6. C

    Orthonormal Set spanning the subspace (polynomials)

    Homework Statement In the linear space of all real polynomials with inner product (x, y) = integral (0 to 1)(x(t)y(t))dt, let xn(t) = tn for n = 0, 1, 2,... Prove that the functions y0(t) = 1, y1(t) = sqrt(3)(2t-1), and y2 = sqrt(5)(6t2-6t+1) form an orthonormal set spanning the same subspace...
  7. C

    Finding an orthonormal basis for a subspace

    Homework Statement Find an orthonormal basis for the subspace of V4 spanned by the given vectors. x1 = (1, 1, 0, 1) x2 = (1, 0, 2, 1) x3 = (1, 2, -2, 1) Homework Equations Gram-Schmidt Process The Attempt at a Solution I have used the Gram-Schmidt process but seem to be running into trouble...
  8. A

    MHB Determining if a set is a subspace.

    Hey there guys, its AngrySnorlax here again with another problem. I posted here before when I was having an issue and the responses I got were extremely helpful because there was a specific step that I just could not grasp that was explained to me and I am hoping that is the same situation here...
  9. S

    "Span of a Subspace - Does it Equal x?

    Homework Statement If x is a subspace of V so, span(x)=x Homework Equations span(x)=x The Attempt at a Solution If x is a subspace so, for any "a", "b" in x: a+b∈x and (c1)*a∈x So a linear combination of x belongs to x but is equal to x?
  10. T

    Is the Zero Vector in the Linear Subspace H?

    Lets say H = [3a+b, 4, a-5b] where a and b are any scalars. This not a vector space because Zero vector is not in H. I don't get what it means by zero vector is not in H? Can't you just multiply the vectors by zero and get a zero vector? I am confused.
  11. N

    Set of vectors whose coordinates are integer (is a subspace?)

    Homework Statement For a set of vectors in R3, is the set of vectors all of whose coordinates are integers a subspace?The Attempt at a Solution I do not exactly understand if I should be looking for a violation or a universal proof. If x,y, z \in Z then x,y,z can be writted as...
  12. N

    Explanation of solution to given question (subspace)

    Homework Statement -4x1 +2x2 =2 4x1 -3x2 -2x3 =-3 2x1 -x2 +(k - k2)x3 =-k Find the values of k for which the system has 1) unique solution, 2) infinitely many solution and 3) no solution The Attempt at a Solution In REF: the matrix is -4 2 0 | 2 0 -1 -2 | -1 0 0 k(1-k)...
  13. N

    Basis for Subspace: Find & Check LI

    Homework Statement Find the basis for the subspace 4x+y-3z The Attempt at a Solution I found that the basis is {[1;-4;0],[0;3;1]}. How do I know if it is linearly independent? I know that the mathematical definition of what LI is but how can it be applied to show in this case?
  14. N

    Proving Kernel of T is a Subspace of V

    Homework Statement I've been stuck on this problem for a while, I actually have the answer (found it in my book), but I'm having trouble getting my head around the concept. The question is: Given a linear transformation T:V->W prove that kernel(T) : {vεV : TV=0W} is a subspace of V...
  15. sheldonrocks97

    An Example of a 2-Dimensional Subspace of C[0,1]

    Homework Statement Give an example of show that no such example exists. A two dimensional subspace of C[0,1] Homework Equations None that I know of. The Attempt at a Solution I know that C[0,1] is a set of continuous functions but I'm not sure where to go after that.
  16. L

    Orthochronous subspace of Lorentz group.

    In a Lorentz group we say there is a proper orthochronous subspace. How can I prove that the product of two orthchronous Lorentz matrices is orthochronous? Thanks. Would appreciate clear proofs.
  17. C

    Vector S, dimension of subspace Span(S)?

    Homework Statement Consider the set of vectors S= {a1,a2,a3,a4} where a1= (6,4,1,-1,2) a2 = (1,0,2,3,-4) a3= (1,4,-9,-16,22) a4= (7,1,0,-1,3) Find the dimension of the subspace Span(S)? Find a set of vectors in S that forms basis of Span(S)? Homework Equations dimension of V = n in Rn...
  18. C

    Dimension of set S, subspace of R3?

    Homework Statement Determine whether set S = {2a,-4a+5b,4b| aε R ^ bε R} is a subspace of R3? If it is a subspace of R3, find the dimension? Homework Equations dimension= n if it forms the basis of Rn, meaning that its linear independent and span(S) = V The Attempt at a...
  19. M

    Find a Basis B for the subspace

    Homework Statement Let V be the subspace of R3 defined by V={(x,y,z)l2x-3y+6z=0} Find a basis B for the subspace. Homework Equations The Attempt at a Solution First I broke apart the equation such that: [[x,y,z]] = [[3/2s-3t, s, t]] = s [[3/2, 1 ,0]] +t[[-3, 0, 1]]...
  20. N

    Test for subspace (is 1 condition sufficient?)

    Homework Statement U= { (x1, x2, x3, x4) | x1 x3 ≥ -5 } The Attempt at a Solution Let x = (1,2,3,4) and y = (1,2,3,4) x+ y = (2,4,6,8) x1x3 = 2x6 = 12 12 >-5 so closure by addition is fulfilled. I've been hearing contradicting information-some state that any 1 test of...
  21. AwesomeTrains

    Finding the Dimension of the Union of Subspaces

    Evening everyone, I have a problem with addition of subspaces. Homework Statement I have to find the dimension of U and dim(V), of the union dim(U+V) and of dim(U\capV) U is spanned by \begin{align} \begin{pmatrix} 1 \\ -2 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\...
  22. N

    Parametric equation in subspace

    Homework Statement The following describes a subset S of R3, you are asked to decide if the subset is a subspace of R3. x = 1-4t y = -2-t z = -2-t The Attempt at a SolutionR3 = {(1-4t, -2-t, -2-t) | t element of all Real number}If S is a subset, at least one must be true. 1) must contain...
  23. N

    Subset & Subspace Homework: Closed Under Vector Addition & Scalar Multiplication

    Homework Statement a) Find a set of vectors in R2 that is closed under vector addition but not under scalar multiplication Find a set of vectors closed under scalar multiplication but not closed under vector addition. The Attempt at a Solution a) Let S be a set of vectors in R2...
  24. N

    Show that a line in R2 is a subspace

    Homework Statement Show that a line in R2 is a subspace if and only if it passes through the origin (0,0) The Attempt at a Solution Let A set of vectors be the subset of the vector space R2. What does it implies in context of this problem if it passes through the origin (0,0)? Does it means...
  25. N

    Proving Subset and Subspace Properties | V is a Subspace of Rn

    Homework Statement Show that if V is a subspace of R n, then V must contain the zero vector. The Attempt at a Solution If a set V of vectors is a subspace of Rn, then, V must contain the zero vector, must be closed under addition, and, closed under scalar multiplication. Let u =...
  26. N

    S, T and U are all subspaces of R4. Subspace & Subset Homework in R4

    Homework Statement S = { (x1, x2, x3, x4) | 4 x1 + x3 = 3 + 6 x2 + x4 } T = { (x1, x2, x3, x4) | x1 + x3 is an integer } U = { (x1, x2, x3, x4) | x1 x3 ≥ -5 } The Attempt at a Solution a) Which of these subsets contain the zero vector 0 = (0, 0, 0, 0) ? S = (x1,x2,x3,x4) =...
  27. N

    Subsets & Subspaces: Determine 0 Vector & R4

    Homework Statement a)Which of these subsets contain the zero vector 0 = (0, 0, 0, 0) ? b)Which of these subsets are subspaces of R4 ?S = { (x1, x2, x3, x4) | x4 = -6 - 5 x1 } T = { (x1, x2, x3, x4) | x4 is an integer } U = { (x1, x2, x3, x4) | x1 + x4 ≤ -6 } The Attempt at a Solution If a...
  28. Chacabucogod

    Shilov's Linear Algebra Subspace Question

    Hi, I'm reading Shilov's linear algebra and in part 2.44 he talks about linear independent vectors in a subspace L which is a subset of space K( he refers to it as K over L). I don't understand why he says that a linear combination of vectors of the subspace L and vectors of the subspace K...
  29. N

    Show that a line in R2 is a subspace problem

    Homework Statement Show that a line in R2 is a subspace if and only if it passes through the origin (0,0) The Attempt at a Solution S={(x,y)| (x,y) =(0,0)} Or S = {(x,y)|x=y} Am I setting up the problem correctly?
  30. B

    MHB Invariant subspace for normal operators

    I have proved the spectral theorem for a normal operator T on an infinite dimensional hilbert space, and am now trying to deduce that T has non-trivial invariant susbspaces. Case 1: If the spectrum of T consists of a single point: My book says that if this is the case then the set of continuous...
  31. G

    Subspace of 2D space of physics

    Please excuse me for my less knowledge. I always tried to physically visualise mathematics facts. My first question is " Is 1D space of physics a subspace of 2D space of physics and so on... So in this way our 3D space is a subspace of 4 D space(spacetime). Can I imagine applying all properties...
  32. O

    Determine if S is a subspace of V

    My answers aren't all correct and I am not sure why.. Problem: Determine whether the given set S is a subspace of the vector space V. A. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying...
  33. ChrisVer

    Fully reducible rep-invariant subspace

    How could I show that a representation D is fully reducible if and only if for every invariant subspace V_{1} \in V then also V_{1}^{T} (meaning orthogonal to V1) is also invariant? http://www.crystallography.fr/mathcryst/pdf/nancy2010/Souvignier_irrep_syllabus.pdf (Lemma 1.6.4) in fact I...
  34. N

    Finding a Basis for P2 Subspace with p'(5)=0

    Homework Statement The problem asks, find a basis for the P2 subspace that consists of polynomials, p(x) such that p'(5)=0. The Attempt at a Solution I know that a set of vectors is a basis if it's linear independent and spans the vector space. So I let p(x) = ax2 + bx +c ...
  35. I

    Subspace Verification in R4: Homework Question & Solution

    Homework Statement In each case below, either show that the set W is a subspace of R4 or give a counterexample to show it is not. a) ##W=\{(x_{1},x_{2},x_{3},x_{4})|x_{4}=x_{1}+x_{3}\}## b) ##W=\{(x_{1},x_{2},x_{3},x_{4})|x_{1}-x_{2}=1\}## Homework Equations The Attempt at a...
  36. Sudharaka

    MHB Distance Between a Subspace and a Vector

    Hi everyone, :) I just want to confirm my answer to this question. Question: Find the distance between a vector \(v\) and a subspace \(U\) in a Euclidean space \(V\). Answer: Here what we have to find essentially, is the length of the projection of \(v\) to the orthogonal compliment...
  37. T

    Find T-cyclic subspace, minimal polynomials, eigenvalues, eigenvectors

    Homework Statement Let T: R^6 -> R^6 be the linear operator defined by the following matrix(with respect to the standard basis of R^6): (0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 ) a) Find the T-cyclic subspace generated by each standard basis vector...
  38. 1

    Two linear transformations agree, subspace

    I've been up way too long, so pardon me if this doesn't make sense, but.. Let V and W be vector spaces. Let T and U be linear transformations from V to W. Consider the set of all x in V such that T(x) = U(x) 1.) I think that this is a subspace of V. 2.) Can I say anything about its dimension...
  39. P

    Which of the subsets of R^3 is a subspace of R^3.

    1. Which of the subsets of R3 is a subspace of R3. a) W = {(x,y,z): x + y + z = 0} b) W = {(x,y,z): x + y + z = 1} I was wondering if my answer for A is correct. Homework Equations 3. A) W = {(x,y,z): x + y + z = 0} Since, x + y + z = 0. Then, the values for all the...
  40. P

    Determine whether W is a subspace of the vector space

    1. Determine whether W is a subspace of the vector space. W = {(x,y,z): x ≥ 0}, V = R3 I am not sure if I am doing this right. 2. Test for subspace. Let these conditions hold. 1. nonempty 2. closed under addition 3. closed under scalar multiplication 3. Testing for...
  41. T

    MHB Prove G is a Subspace of V ⊕ V and Quotient Space (V ⊕ V)/G Isomorphic to V

    Prove that G is a subspace of V ⊕ V and the quotient space (V ⊕ V) / G is isomorphic to V. Let $V$ be a vector space over $\Bbb{F}$, and let $T : V \rightarrow V$ be a linear operator on $V$. Let $G$ be the subset of $V \oplus V$ consisting of all ordered pairs $(x, T(x))$ for $x$ in $V$. I...
  42. S

    MHB Distance from a vector to a subspace

    Hello everyone Here is the question Find the distance from a vector $v=(2,4,0,-1)$ to the subspace $U\subset R^4$ given by the following system of linear equations: $2x_1+2x_2+x_3+x_4=0$ $2x_1+4x_2+2x_3+4x_4=0$ do I need to find find a point $a$ in the subspace $U$ and write the vector $a-v$...
  43. T

    MHB Prove that the quotient space R^n / U is isomorphic to the subspace W

    Let A be an m x n matrix with entries in R. Let T_A : R^n -> R^m be the linear map T_A(X) = A_X. Let U be the solution set of the homogeneous linear system A_X = O. Let W be the set of all vectors Y such that Y = A_X for some X in R^n. I don't really know what I'm supposed to do here, any help...
  44. M

    Connected components of a metric subspace

    Homework Statement . Consider the subspace ##U## of the metric space ##(C[0,1],d_∞)## defined as ##U=\{f \in C[0,1] : f(x)≠0 \forall x \in [0,1] \}##. Prove that ##U## is open and find its connected components. The attempt at a solution. First I've proved that ##U## is open. I want to...
  45. J

    Is W a Subspace of R3? Understanding its Characteristics

    I want to know why this subset W is a subspace of R3. W is defined as: | x+2y+3z | | 4x+5y+6z | | 7x+8y+9z | I know the possible subspaces of R3 are the origin itself, lines through the origin, and planes through the origin. Would W be a subspace of R3 simply because there would be...
  46. D

    Proving that the orthogonal subspace is invariant

    Hi guys, I couldn't fit it all into the title, so here's what I'm trying to do. Basically, I have a unitary representation V. There is a subspace of this, W, which is invariant if I act on it with any map D(g). How do I prove that the orthogonal subspace W^{\bot} is also an invariant subspace...
  47. B

    Write the subspace spanned by vectors as a kernel of a matrix.

    Hi Lets say I have a vectorspace in Rn, that is called V. V = span{v1,v2,... vk} Is it then possible to create an m*n matrix A, whose kernel is V. That is Ax = 0, x is a sollution if and only if x is an element of V. Also if this is possible, I imagine that k may not b equal to m?
  48. Petrus

    MHB Find a Basis for Subspace in P_3(\mathbb{R})

    Hello, Find a basis for subspace in P_3(\mathbb{R}) that containrar polynomial 1+x, -1+x, 2x Also the hole ker T there T: P_3(\mathbb{R})-> P_3(\mathbb{R}) defines of T(a+bx+cx^2+dx^3)=(a+b)x+(c+d)x^2 I am unsure how to handle with that ker.. I am aware that My bas determinant \neq0 well I did...
  49. 1

    Linear Algebra - Field Subspace

    Homework Statement 1. Let X be a set and F a Field, and consider the vector space F(X; F) of functions from X to F. For a subset Y\subseteq X, show that the set U = {f \in F(X; F) : f |Y = 0 } is a subspace of F(X; F). NB: the expression \f |Y = 0" means that f(y) = 0 whenever y \in Y...
  50. P

    MHB Proof of Eigenvector Existence for Linear Maps on Finite-Dimensional Spaces

    From wikipedia I read that every linear map from T:V->V, where V is finite dimensional and dim(V) > 1 has an eigenvector. What is the proof ?
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