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. D

    Basis of sum/union of subspaces

    Homework Statement I need to find the basis of W1 + W2 and W1 intersect W2 (It's part of a larger homework problem that I know how to do, but I am stuck on the trivial step...per usual) W_1 = \left(\begin{array}{c c} x & -x \\ y & z \end{array}\right) W_2 = \left(\begin{array}{c c} a...
  2. D

    Finding Basis for Subspaces of R4

    i am given 2 subspaces of R4 W=sp{(a-b,a+2b,a,b)|a,b\inR} U=sp{(1,0,1,1)(-6,8,-3,-2)} and am asked to find: a homogenic system for W- system for a vector (x,y,z,t) belonging to W i see the basis for W is : a(1,1,1,0)+b(-1,2,0,1),, i put these vectors into an extended matrix with (x y z...
  3. T

    Comparing subspaces of a linear operator

    Statement Let S be a linear operator S: U-> U on a finite dimensional vector space U. Prove that Ker(S) = Ker(S^2) if and only if Im(S) = Im(S^2) So, I'm really not sure about how to prove this properly. I have a few ideas, but this one seemed to make sens intuitively to me. So, I'm...
  4. D

    Subspaces of R4: W & U & Basis Calculation

    in this question i am given 2 subspaces of R4 W={(x1,x2,x3,x4)\inR4|x1-x4=0} U=sp{(1,2,0,-1),(2,3,1,-1),(1,-1,1,-1)} and am asked to find 1] a basis and dimention for W 2] a basis for W+U 3] a basis for W\capU 1] since i only have limitations on x1 and x4 i call x2=t x3=s...
  5. J

    Is the Intersection of Subspaces a Subspace?

    Homework Statement Let H and K be subspaces of a vector space V. Prove that the intersection K\cap H is a subspace of V. Homework Equations The Attempt at a Solution This, I have absolutely NO idea how and where to start. Are there any axioms which can be used to prove this?
  6. M

    Understanding the Addition of Subspaces in Linear Algebra

    What exactly is the "addition of subspaces?" It is obviously not the same as the "union of subspaces," since the union of subspaces A and B in V is a subspace of V only if A is contained in B (or B is contained in A).
  7. M

    Linear Algebra - adding subspaces

    1. Suppose that U is a subspace of V. What is U+U? 2. Homework Equations : There's a theorem that states: Suppose that A and B are subspaces of V. Then V is the direct sum of A and B (written as A [plus with a circle around it] B) if and only if: 1) V=A+B (meaning, the two...
  8. K

    Vector Spaces & Subspaces, Linear Algebra

    Homework Statement Let V be a vector space and U a subspace of V . For a given x ∈ V , define T= {x + u | u ∈ U }. Show that T is a subspace of V if and only if x ∈ U . Homework Equations Subspace Test: 1: The 0 vector of V is included in T. 2: T is closed under vector addition 3...
  9. S

    Minimal Invariant Subspaces: The Role of Orthogonal Linear Transformations

    I have a question about this theorem. Let V be an n-dimensional inner product space, and let T:V-->V be an orthogonal linear transformation. Let S be a minimal invariant subspace under T. Then S is one dimensional or two dimensional. I understand what this theorem says and I follow the...
  10. D

    Find Basis for Subspaces & Dimension of W,U,W+U,W\capU

    i am given these 2 groups W=sp{(1 0 2 0) (1 1 1 1) (1 0 0 0)} U=sp{(1 0 1 1) (1 2 1 2) (0 0 1 0)} and am asked to find a basis for each one and their dimention a basis for W+U a basis for W\capU ----------------------------------------------- for the basis i found that they are...
  11. J

    Proving Subspaces of R^m: Linear Combinations and Vector Forms

    Homework Statement Show that the following are subspaces of R^m : (a) The set of all linear combinations of the vectors (1,0,1,0) and (0,1,0,1) (of R^4) (b) The set of all vectors of the form (a,b,a-b,a+b) of R^4 Homework Equations The Attempt at a Solution (a) If...
  12. W

    Decomposition of a complex vector space into 2 T-invariant subspaces

    Homework Statement Suppose V is a complex vector space and T \in L(V). Prove that there does not exist a direct sum decomposition of V into two proper subspaces invariant under T if and only if the minimal polynomial of T is of the form (z - \lambda)^{dim V} for some \lambda \in C. Homework...
  13. W

    S^n not a mapping cylinder. S^n and homeom. subspaces

    Hi, everyone: I have been trying to show this using the following: Given f: Y-->X IF S^n ~ Y_f(x) , then S^n deformation-retracts to Y , and ( not sure of this) also is homeomorphic to Y (I know Y_f(x) is homotopic to Y ) . But ( so I am branching out into more...
  14. S

    Proving Subsets of Rn: Steve's Questions Answered

    a) {(x1,x2,x3) | x1+x2 ≥ 0} b) {x∈R3 |proj(1,1,1) (x) ∈ Sp({(1,1,1)})} Prove the set is or is not a subset of R n I have no idea how to solve this. Our textbook gives NO examples of how to prove these Please help me get started, a related example would be great too. :) Thanks-...
  15. W

    Linear Algebra and Matrices, Subspaces, basis

    Homework Statement I'm unclear about this statement being wrong or not: if C is an x-dimensional subspace of Rn, then a linearly independent set of x vectors in C is a basis for C The Attempt at a Solution I think that it must be a basis since it has independent vectors and it is in x...
  16. T

    Subspace & Basis: Proving A is a Subspace of R^3

    Homework Statement Let u be a vector where u = [4 3 1]. Let A be the set of all vectors orthogonal to u. Show that A is subspace of R^3. Then find the basis for A. Homework Equations The Attempt at a Solution For showing that A is a subspace... Zero vector is in A because...
  17. F

    Intersection of Subspaces

    Let H and K be subspaces of a vector space V. Prove that the intersection K and H is a subspace of V. Intuitively I can see that this is true... Both H and K must be closed under vector addition and scalar multiplication so there intersection must also be closed under both those. How do i...
  18. H

    Dimensions of subspaces

    If V and W are 2-dimensional subspaces of R4 , what are the possible dimensions of the subspace V intersection W? I am new to subspaces, so I have no clue to this question. Help guys! Options: (A) 1 only (B) 2 only (C) 0 and 1 only (D) 0, 1, and 2 only (E) 0, 1, 2, 3, and 4
  19. I

    Proving Subspaces in Linear Algebra

    Homework Statement Let S be a nonempty set and F a field. Prove that for any s_0 \in S, {f \in K(S,F): f(s_0) = 0}, is a subspace of K(S,F). K here is supposed to be a scripted F. Homework Equations The Attempt at a Solution I don't know how to approach this problem. I know the three...
  20. J

    Invariant subspaces under linear operators

    Homework Statement Prove or give a counterexample: If U is a subspace of V that is invariant under every operator on V, then U = {0} or U = V.Homework Equations U is invariant under a linear operator T if u in U implies T(u) is in U.The Attempt at a Solution Assume {0} does not equal U does not...
  21. M

    Solving U, V, W Subspaces Problem

    Hi, I have thius problem to solve. Please, help me! 1. Prove or disprove if U, V, W are subspaces of V for which U (dir sum) W = V (dir sum) W then U=V Thank you in advance!
  22. E

    Union of Proper Subspaces Problem

    Homework Statement Let V be a vector space over an infinite field. Prove that V is not the union of finitely many proper subspaces of V. The attempt at a solution Suppose V is the union of the proper subspaces U1, ..., Un. Let ui be a vector not in Ui. If u1 + ... + un is in the union...
  23. D

    Exploring Subspaces of $\mathbb{R}^n$: Is S a Subspace?

    Let \vec{u},\vec{v},\vec{w} be fixed vectors in Rn. Define S to be the set of all vectors in Rn which are linear combinations of the form k_1 \vec{u}+k_2 \vec{v}+3 \vec{w}, where k_1,k_2 \in R. Is S a subspace of Rn? Im a little stuck with this one. I've tried defining two vectors...
  24. M

    Triangular matricies and subspaces

    hello again I was asked if the set of all uppertriangular nxn matricies are a subspace of Mnn, how would you check if it has a zero vector and closed under addition and multiplication ? and why did they ask for the upper triangular matrix instead of the lower one? or either
  25. rocomath

    What are the types of subspaces in R^4?

    There are more answers to this problem, but I'm not sure how to approach it. The subspaces of R^3 are planes, lines, R^3 itself, or Z containing only (0,0,0,0). b. Describe the five type of subspaces of R^4 i. lines thru (0,0,0,0) ii. zero (0,0,0,0) iii. planes thru (0,0,0,0)...
  26. S

    Linear Algebra - Topic: Basis & Subspaces

    I have a few questions here, my main problem is not understanding the notations used, hence not understanding the questions. Homework Statement 1. Do the vectors j_{1}= (1,0,-1,2) and j_{2}= (0,1,1,2) form a basis for the space W = {(a,b,c,d) l a - b + c = 0, -2a - 2b + d =0} ? 2...
  27. L

    What is the relationship between subspaces V and W if V is contained in W?

    subspaces and dimension! Consider two subspaces V and W of R^n ,where V is contained in W. Why is dim(V)<= dim(W)...? "<=" less than or equal to
  28. P

    Subspaces of R2 and R3: Understanding Dimensions of Real Vector Spaces

    So I'm considering dimensions of real vector spaces. I found myself thinking about the following: So for the vector space R2 there are the following possible subspaces: 1. {0} 2. R2 3. All the lines through the origin. Then I considered R3. For the vector space R3 there are the...
  29. C

    Linear Algebra: 4 Fundamental Subspaces

    Homework Statement Without computing A, find the bases for the 4 fundamental subspaces. [1 0 0][1 2 3 4] [6 1 0][0 1 2 3]=A=LU [9 8 1][0 0 1 2] Homework Equations N/A The Attempt at a Solution There was an "example" in the book. It just showed the answers. It was: [1 0...
  30. K

    Vector spaces, subspaces, subsets, intersections

    Homework Statement Let V be a vector space over a field F and let X, Y and Z be a subspaces of V such that X\subseteqY. Show that Y\cap(X+Z) = X + (Y\capZ). (Hint. Show that every element of the LHS is contained on the RHS and vice versa.) Homework Equations The Attempt at a...
  31. K

    Are These Subsets of R^R Subspaces?

    Homework Statement Which of the following subsets of the vector space R^R of all functions from R to R are subspaces? (proofs or counterexamples required) U:= f \inR^R, f is differentiable and f'(0) = 0 V:= f\inR^R, f is polynomial of the form f=at^2 for some a\inR = There exists a...
  32. S

    Lin. Algebra - Sum of Dim. of Three Subspaces

    Another linear algebra question! What a surprise! :rolleyes: Homework Statement If U1, U2, U3, are subspaces of a finite-dimensional vector space, then show dim(U1 + U2 + U3) = dimU1 + dimU2 + dimU3 - dim(U1 \cap U2) - dim(U1 \cap U3) - dim(U2 \cap U3) + dim(U1 \cap U2 \cap U3) or give a...
  33. S

    Linear Algebra - Vector Subspaces

    Homework Statement Give an example of a nonempty subset U of R2 such that U is closed under scalar multiplication, but U is not a subspace of R2. Homework Equations The Attempt at a Solution I think I have it, but I just want to make sure it's right: Let U = {(x, x + 2)} | x is...
  34. N

    How do I find orthogonal projections on subspaces?

    [SOLVED] Projections on subspaces Homework Statement I have some questions on this topic: 1) If I have an orthonormal basis for a subspace U and I have a vector A, and I want to find the orthogonal projection of A onto U, then I use the expression written here...
  35. M

    What is the Proper Subspace Hierarchy in a Vector Space with Infinite Subsets?

    Homework Statement Let {W_1,W_2,W_3,...} be a collection of proper subspaces of V (i.e. W_i not=V) such that W_i is a subset of W_(i+1) for all i. Prove that U(W_i) (i from 1 to infinity) is a proper subspace of V The Attempt at a Solution I've already proven that U(W_i) is a subspace of...
  36. M

    Vector Spaces, Subsets, and Subspaces

    Homework Statement What is an example of a subset of R^2 which is closed under vector addition and taking additive inverses which is not a subspace of R^2? R, in this question, is the real numbers. Homework Equations I know that, for example, V={(0,0)} is a subset for R^2 that...
  37. M

    Example of a Non-Subspace in R^2 Closed Under Addition and Inverses?

    I don't understand this, can someone help?: What is an example of a subset of R^2 which is closed under vector addition and taking additive inverses which is not a subspace of R^2? R, in this question, is the real numbers. Thanks!
  38. E

    Proving V = U ⊕ W: A Subspaces Problem

    Hi, I've being going over some course material in preparation for my exams and I've come across an exercise which I can't seem to work out. The exercise: Let V be a vector space over a field F and let U and W be two subspaces of V. Suppose V = U + W. Prove that V = U ⊕ W iff {(u,w)∈U×W : u...
  39. Z

    Is the Intersection of Subspaces a Subspace in a Vector Space?

    Hi to all, I really need help fast. How do I solve this question? A solution would be much appreciated. THANKS A MILLION! ======================================================= Let S1 and S2 be the two subspaces in a vector space V. Show that the intersection of S1 and S2 is also...
  40. G

    Subspaces and Basis of vector spaces

    I am totally lost on the following questions. What does exhibit mean? 1) Show that the given set H is a subspace of ℜ^3 by finding a matrix A such that N(A) = H (in this case, N(A) represents the null space of A). 2) Exhibit a basis for the vector space H. a b {for all R^3...
  41. M

    Linear Algebra: Invariant Subspaces

    Homework Statement Prove or give a counterexample: If U is a subspace of V that is invariant under every operator on V, then U = {0} or U = V. Assume that V is finite dimensional. The attempt at a solution I really think that I should be able to produce a counterexample, however...
  42. M

    Finding Subspaces that Satisfy Specific Intersections in \mathbb{R}^3

    Homework Statement Find subspaces A, B, and C of \mathbb{R}^3 so that A \cap B \cap C \ne \{\vec{0}\} and (A + B) \cap C \ne A \cap C + B \cap C. You can specify a subspace by the form A = span\{\vec{e}_1, \vec{e}_2\}.Homework Equations A + B is the set of all vectors in \mathbb{R}^3 of the...
  43. M

    Subspace Addition: Understanding the Union of A and B

    I just wanted to know if subspace A + subspace B is the same as the "union of A and B".
  44. M

    Exploring Vector Subspaces and the Zero Vector

    Hello Everyone. In a thread in this forum relating to a problem on Subspaces I read that as long as a Vector SubSpace is closed under addition and multiplication we always have the zero vector. I can see that we can always get the zero vector but do we not have to define a zero vector first...
  45. R

    Unitary, invariant subspaces

    Homework Statement Let U be a unitary operator on an inner product space V, and let W be a finite-dimensional U-invariant subspace of V. Prove that (a) U(W) = W (b) the orthogonal complement of W is U-invariant (for ease of writing let the orthogonal complement of W be represented by...
  46. N

    Is R={(x,y,z,w) belongs to R^4; x+y+sz-w=s^2-2s} a subspace of R^4?

    I came across a problem in linear algebra asking me to find for which values of s the following set: R={(x,y,z,w) belongs to R^4; x+y+sz-w=s^2-2s} is a subspace of R^4 with respect to the usual additoin and scalar multiplication. Any idea to solve this question? Thanks in advance
  47. R

    Chapter07.pdfCan Killing Vectors Derive the Schwarzschild Metric?

    "Is it possible to derive the Schwarzschild metric from Killing vectors, thus saving all that work with the Ricci tensor etc."
  48. Q

    Thank you!Is my understanding of T-invariant subspaces correct?

    Hello, can anyone tell me if I understand this right? :rolleyes: I have a t-invariant subspace with basis B, and I extend the basis B to be a basis B' for the entire vector space by adding L.I. vectors to it. Then I put B under a linear transformation, T:V --> V, and I will get a set of...
  49. Q

    How to find bases for subspaces

    Homework Statement I'm given a subspace in F^5 (not sure how to note that online) and asked to find a basis and dimension for it. I know it should be really easy, but ... Homework Equations We're given subspace W1 = {a1,a2,a3,a4,a5) in F^5: a1-a3-a4=0} . We also know from linear...
  50. S

    Subspaces of P spaces and C[a,b] spaces

    Homework Statement Determine whether the set of polynomials of degree 3 form a subspace of P(4) Homework Equations P(4) = c_3 x^3 + c_2 x^2 + c_1 x + c_0 The Attempt at a Solution \alpha P(4_1) = \alpha c_3 x^3 + \alpha c_2 x^2 + \alpha c_1 x + \alpha c_0 This just scales the...
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