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. W

    Can a Complex Matrix X Solve the Column Subspace Problem?

    Let A_i (i=1,...,k) be a nonsingular complex matrix which size is M by M. The question is how to find a complex matrix X which size is M by N such that: span(A_1*X)=...=span(A_k*X) (I guess that there must be relations between M,N and k when nontrival solution exists. ) ask: 1)if non...
  2. S

    Basis for a subspace in R^4

    Homework Statement Find a basis for F=\left\{(x,y,z,w): -x+y+2z-w=0\right\}The Attempt at a Solution So this looks like a plane to me, but I find 4-d space confusing, so that might be wrong. It does have the form \mathbf{x}^T\mathbf{n}=0, so that's kind of where I'm getting the idea that it's...
  3. G

    Proving that something is a subspace of all the infinite sequences

    Homework Statement let V be the vector space consisting of all infinite real sequences. Show that the subset W consisting of all such sequences with only finitely many non-0 entities is a subspace of VThe Attempt at a Solution Ok so i have to show 1.Closure under Addition,2. Closure under...
  4. S

    Linear Algebra subspace troubles

    Homework Statement Let V be a finite dimensional subspace. Let V\supseteqU1\supseteqU2\supseteq...\supseteqUk. Show that there exists k such that Uk=Uk+1=...=Un=...Homework Equations We were also told to assume none of the subspaces are zero dimensional, and to think about how the dimensions...
  5. P

    Finding a Basis for Perpendicular Vectors in R4

    Homework Statement Find a basis for each of these subspaces of R4 All vectors that are perpendicular to (1,1,0,0) and (1,0,1,1) 2. The attempt at a solution I'm not sure how to approach this question. The only thing I can think of is that a vector that would be perpendicular to both would be...
  6. F

    Topology: is this an open cover of an unbounded subspace of a metric space?

    Homework Statement Suppose A is an unbounded subspace of a metric space (X,d) (where d is the metric on X). Fix a point b in A let B(b,k)={a in X s.t d(b,a)<k where k>0 is a natural number}. Let A^B(b,k) denote the intersection of the subspace A with the set B(b,k). Then the...
  7. S

    The Perpendicular Subspace of R^n: What is it and How is it Defined?

    Homework Statement The Attempt at a Solution The terminology in this question confuses me into what I am actually trying to solve. It seems to me that S-perp would naturally be a subspace of real column vectors based on the fact that we specify that S\neq0. It goes on to mention...
  8. J

    Is the Set of Functions with a Zero Integral a Subspace of C[a,b]?

    Homework Statement Determine whether or not the given set is a subspace of the indicated vector space: Functions f such that [integral from a to b]f(x)dx = 0; C[a,b] (not sure how to do the coding for integrals) Homework Equations to be a subspace it must follow these axioms: (i) if x and y...
  9. H

    Proving Vector Subspace: Field, Matrix, and Basis Properties

    Hey guys, I've got another problem I could use some assistance with. "In this problem we suppose that F is a field, A is an m by n matrix over F and that W is a subspace of Fm. (a) Show that U = {v \in Fn: Av \in W} is a subspace of Fn. (b) Now suppose that m = n and A is invertible, and that B...
  10. H

    Subspace of even and odd functions

    Hey guys. I came upon this problem in the professor's recommended problems, and I have no idea how to solve it. After spending an hour on it, I've got nothing. Any suggestions would be much appreciated! A function f : R -> R is called even just in case f(-r) = f(r) for every real number r. f...
  11. Talisman

    Is the Symmetric Difference of Finite-Dimensional Subspaces Well-Defined?

    Suppose U and V are finite-dimensional subspaces of some finite-dimensional space W (over the field Q in my actual case, but probably irrelevant) I've got a subspace in mind that I can't quite define well. I'll start with a concrete example: Consider R^3 with orthonormal basis {x, y, z}. Let U...
  12. A

    How to find a basis of a subspace V = {(x1; x2;….; xn) | Σni=1 xi=0}

    Given V = {(x1; x2;….; xn) | Σni=1 xi=0} (sum of vectors is equal to zero) be a subspace of Rn. How can we find a basis of V such that for each vector {(x1; x2;….; xn) in the basis Σi=1n x2i=1 ( i.e. sum of squares is equal to 1).
  13. I

    Subspace of Normed Vector Space

    Let X be a normed vector space. If C is a closed subspace x is a point in X not in C, show that the set C+Fx is closed. (F is the underlying field of the vector space).
  14. P

    Prove the sum of two subspaces is also a subspace.

    Homework Statement Hi I'm trying to prove that the sum of two subspaces U and W is also a subspace. Homework Equations U is a subspace of V if U is also a vector space and it contains the additive identity, is closed under addition, and closed under scalar multiplication. The definition of...
  15. P

    Confused about the proof of U = U + U, where U is a subspace

    Homework Statement Hi I need help understanding a proof. This is my first time in a pure math class, so proofs of this type are a little weird to me. If U is a subspace of the vectorspace V, what is U+U? Homework Equations The proof: (v_{1}+v_{2})\in U+U As v_{1},v_{2}\in...
  16. G

    Determining if W is a Subspace of R3

    Homework Statement For the following subset W of R3 determine whether or not W is a subspace of R3. If the subset is not a subspace give a specific example to indicate why it is not a subspace. ii.) W = {(x,y,z): 2x + y + 3z = 0 The Attempt at a Solution I know how to do this...
  17. B

    Proving Vector Subspaces with Linear Transformations | Homework Help

    Homework Statement Let V and W be vector spaces over F and T:V \rightarrow W a linear transformation. Prove that ker(T):={\epsilon V\mid T()=0_{v}} is a vector subspace of V Homework Equations The Attempt at a Solution Is it all right just to state the trivial solution. ie...
  18. U

    Reconstructing operator matrix from subspace samples

    Hi, I wonder if there is some agreed-upon best way to reconstruct the matrix of a positive definite operator A using "sampling" (like in tomography). More in detail I want to do this: I have many small sets of basis functions. The sets are in general not orthogonal. I compute matrix elements...
  19. R

    Dense subspace of 1st category

    There's a problem from Rudin's Functional Analysis where I need to show something is a dense subspace of 1st category. But I thought that it was the definition of dense that its closure is the whole space. Hence the closure doesn't have empty interior. So the dense subspace can't be 1st...
  20. J

    Proving Subspace of a Matrix in R2

    Hi. if anybody can help me Let S be the matrix a b c d with a constrain a = -2d, b= 3c -d Prove that S is a subspace of R2?
  21. M

    Subspace Intersection Problem: Proving W1 and W2 in R^n

    Homework Statement Let W1 and W2 be two subspaces of R^n. Prove that their intersection is also a subspace. Homework Equations The Attempt at a Solution I know that in R^2 and R^3 the intersection would be the origin, which would be the zero vector, which would be a subspace...
  22. Saladsamurai

    Prove that the intersection of subspaces is subspace

    Homework Statement Prove that the intersection of any collection of subspaces of V is a subspace of V. Okay, so I had to look up on wiki what an intersection is. To my understanding, it is basically the 'place' where sets or spaces 'overlap.' I am not sure how to construct the problem...
  23. Saladsamurai

    Part 2Find a Subset that is not a Subspace

    Here we go again:biggrin: Give an Example of a subset U of R^2 that is closed under scalar multiplication, but is not a subspace of R^2 I am thinking let U={(x1,x2) : x1=-x2} If x=(x1,x2) and y=(y1, y2) then x+y= (-x2-y2), x2+y2) = (-(x2+y2), x2+y2) okaaayy so that does not work...
  24. Saladsamurai

    Find a Subset that is not a Subspace

    Homework Statement Give an Example of a subset U of R2 that is closed under addition and under taking additive inverses (i.e., -u in U whenever u in U), but is not a subspace of R2 Okay, I know that this problem is not hard, but I just need a hint. I don't want to just start arbitrarily...
  25. Saladsamurai

    Verify Subset is a Subspace

    Okay then. I just read the section of Axler on subspaces. It says that if U is a subset of V, then to check that U is a subspace of V we need only check that U satisfies the following: additive identity 0\in U closed under addition u,v\in U\text{ implies }u+v\,\in\,U closed under scalar...
  26. T

    Convergence of Rational Sequences in Subspaces

    Homework Statement From Introduction to Topology by Bert Mendelson, Chapter 2.7, Exercise 8: Consider the subspace (Q, d_Q) (the rational numbers) of (R, d). Let a1, a2, ... be a sequence of rational numbers such that \lim_{n} a_n = \sqrt{2}. Does the sequence converge when considered...
  27. J

    Linear Algebra: dimension of subspace question

    Homework Statement Find an example of subspaces W1 and W2 in R^3 with dimensions m and n, where m>n>0, such that dim(intersection of W1 and W2)= n Homework Equations dim(W1+W2)= dim(W1) + dim(W2)-dim(intersection of W1 and W2) The Attempt at a Solution Well what I know...
  28. J

    Find Basis for Subspace of R^4 Generated by z

    Homework Statement Suppose T is a linear operator on R^4 such that T(a,b,c,d) = (a + b, b - c, a + c, a + d). Find a basis for the T-cyclic subspace of R^4 generated by z = (1, 0, 0, 0) Homework Equations The Attempt at a Solution I found a basis, but I don't think the method I used was the...
  29. D

    Is this set of polynomials a subspace of P2?

    Homework Statement Let S denote the collection of all polynomials of the form p(t) = (2a - b)t^2 + 3(c - b)t + (a - c), where a,b,c are real numbers. Determine whether or not S is a subspace of P2. The Attempt at a Solution Okay, so I know that in order for S to be a subspace, it must...
  30. M

    Determining if a set is a subspace

    Homework Statement Determine whether or not the set of all functions f such that f(1)+f(-1)=f(5) is a subspace of the vector space F of all functions mapping R into R. Homework Equations The Attempt at a Solution I think that (f(1)+f(-1))+(g(1)+g(-1))=(f+g)(1)+(f+g)(-1)=(f+g)(5)...
  31. J

    Determine if the space is a subspace testing both closure axioms

    Homework Statement determine if the space is a subspace testing both closure axioms. in R^2 the set of vectors (a,b) where ab=0 Homework Equations The Attempt at a Solution i just used the sum and product which are the closure axioms. But at the end how do you tell if the...
  32. S

    Subspace Question: Determine if H is a Subspace in Vector Space

    Homework Statement Determine if the sets are a subspace of the real vector space: Prof is kinda hard to hear and doesn't explain stuff that well, can I get some help with this one? Homework Equations H = {[a,b,c,d] exist in 4-space| 4a+2b-8c+2d = 3a-5b+6d = b-6c-2d = 0} H =...
  33. F

    Squaring a vector and subspace axioms

    If a problem I'm doing asks to find V2 where V is a vector is it simply the dot product of the vector, or the cross product? The question: Which of the following sets of vectors v = {v1,...,vn} in Rn are subspaces of Rn (n>=3) iii) All v such that V2=V12 He proved it by saying...
  34. T

    Is Q a Subspace of R2/R3?

    Hey, this isn't for homework per se, but if anyone could lend me a hand figuring this out I'd appreciate it a lot! Homework Statement Determine whether Q is a subspace of R2/R3 in the following cases: Homework Equations Q = \{\left v = \left( v1, v2, 0 \right) | v1,v2 \in R...
  35. B

    Orthogonal complement of a subspace

    Homework Statement Let W be the subspace spanned by the given column vectors. Find a basis for W perp. w1= [2 -1 6 3] w2 = [-1 2 -3 -2] w3 = [2 5 6 1] (these should actually be written as column vectors. Homework Equations The Attempt at a Solution So, I...
  36. Y

    Why Set of nxn Matrices w/ Zero Determinant Not Subspace of Mn,n

    Hi, I would like to know why the set of all n*n matrix whose determinant is zero is not a subspace of Mn,n .Can anyone explain the reason for me? Thanks!
  37. P

    Help proving a subset is a subspace

    Homework Statement Prove that the set of all 3-vectors orthogonal to [1, -1, 4] forms a subspace of R^3. Homework Equations Orthogonal means dot product is 0. The Attempt at a Solution I know the vectors in this subspace are of the form [a,b,c] where a - b + 4c = 0. However I...
  38. A

    Tricky subspace & intersection Problem

    Homework Statement I am trying to solve this problem: Let W_1, W_2, W_3 be subspaces of a vector space, V. Prove that W_1 ∩ (W_2 + ( W_1 ∩ W_3)) = (W_1 ∩ W_2) + (W_1 ∩ W_3). Can someone help me show this? I have tried using Dedekind's law, but not sure it that is the way to go. The...
  39. D

    Projection onto subspace along subspace

    Homework Statement Find a projection [matrix] E which projects R2 onto the subspace spanned by (1,-1) along the subspace spanned by (1,2).Homework Equations P = \frac{a a^{T}}{a^{T} a}The Attempt at a Solution Computing P... P = \[ \left( \begin{array}{ccc} \frac{1}{2} & -\frac{1}{2}\\...
  40. S

    Is Span{u1,u2,...,um} a Subspace of R^n?

    I have a problem. Suppose that {u1,u1,...,um} are vectors in R^n. Prove, dircetly that span{u1,u2,...,um} is a subspace of R^n. How would I go by doing this?
  41. R

    Proving U is a Subspace of R3

    Homework Statement Let U={(x,y,z) \in R3 : x=z}. Show that U is a subspace of R3. Homework Equations The Attempt at a Solution U is non-empty it contains the 0 vector: U= {(x,y,z) = (s,t,s), s,t \in R} U={s(1,0,1)+t(0,1,0), s,t \in R} for s,t=0...
  42. R

    Proof of W as Subspace of Rn: Help & Explanation

    I saw this problem in a book, it asks if there are two subspaces of Rn, say U & V and the following condition is true: W={w \in R^n : w=u+v for some u \in U and v \in V} Make a proof/show that W is a subspace of Rn. I think maybe we need to try to somehow prove that the set W is a...
  43. S

    What is the basis for the given subspace in R^5?

    Homework Statement find a basis for the subspace R^5 that consists of all the vectors of the form [(b-c), (d-2b), (4d), (c-2d), (6d+2b)] Homework Equations The Attempt at a Solution the only solution I can think of is e1, e2, e3, e4, e5... I don't think it's that simple...
  44. A

    Find a basis of the subspace W:=A

    Homework Statement find a basis of the subspace W:=A\in M2*2(R) : trace (A)=0 of the vector space M2*2 (R) and hence determine the dimension of W Homework Equations The Attempt at a Solution trace(A) denote the the sum of the diagonal elements of the matrix A=aij do i need to...
  45. X

    Real subspace and not real subspace

    Homework Statement 1) Explain why the set W={(x,y)inside dimension 2; |x|=|y|} is not a real subspace 2) Show that the set V={[a b];a+d=0} is a real subspace of dimension 3 {[c d] } Homework Equations The Attempt at a Solution
  46. B

    Quick question: Is this plane a subspace of R^3?

    Homework Statement Say you have the plane given by equation 4x + 3y + 4z + 4 = 0 This plane is not a subspace of R^3, right? My reasoning is that this plane can't include the origin, but I just need some clarification to make sure that I understand what a subspace is. Thanks...
  47. D

    Open subspace of a compact space topological space

    It is a fact that if X is a compact topoloical space then a closed subspace of X is compact. Is an open subspace G of X also compact? please consider the following and note if i am wrong; proof: Since G is open then the relative topology on G is class {H_i}of open subset of X such that the...
  48. D

    Open subspace of a compact space

    It is a fact that if X is a compact topoloical space then a closed subspace of X is compact. Is an open subspace G of X also compact? please consider the following and note if i am wrong; proof: Since G is open then the relative topology on G is class {H_i}of open subset of X such that the...
  49. K

    Determine whether or not something is a subspace

    I guess this kind of topic should belong here. :| My understanding of the subspace still isn't solid enough, so I want to know what I know so far is at least correct. By definition, a set of vectors S of Rn is called a subspace of Rn iff for all vectors (I will call them x): 1) (x+y) \in S and...
  50. K

    Determine whether or not something is a subspace

    My understanding of the subspace still isn't solid enough, so I want to know what I know so far is at least correct. By definition, a set of vectors S of Rn is called a subspace of Rn iff for all vectors (I will call them x): 1) (x+y) \in S and 2) kx \in S. Also, the solution set of a...
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