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. P

    Sum of two subspaces - question.

    Homework Statement Is it possible to add the following subspaces: W_1 = Sp{(1,0,0)} and W_3 = Sp{(0,1,-1), (0,0,1)}? Homework Equations The Attempt at a Solution Will their sum be: Sp{(1,1,-1),(1,0,1)}?
  2. P

    Are U and W Equal? Solving for Linear Independence in Subspaces

    Hi, How may I determine whether the subspaces U and W are equal to each other?: K is linearly independent wrt V, defined thus: K={v1,v2,v3,v4} subset of V U and W, subspaces of V, are defined thus: U=Sp(K); W=Sp{v1-v2,v2-v3,v3-v4,v4-v1} I am not allowed to use equality between...
  3. W

    Are Quotients by Homeomorphic Subspaces Homeomorphic?

    Hi, All: Let X be any topological space, and let A,B be subspaces of X that are homeomorphic to each other. Does it follow that the quotients X/A and X/B are homeomorphic? I know this is true if A,B are both contractible in X , since we then have X/A ~X ~X/B But I'm not sure...
  4. W

    Linear Algebra Four fundamental subspaces small proof.

    Homework Statement Given A\in Mnxn and A = A2, show that C(A) +N(A) = ℝn. note: C(A) means the column space of A. N(A) means the null space of A Homework Equations These equations were proved in earlier parts of the problem... C(A) = {\vec{x}\in ℝn such that \vec{x} =...
  5. S

    Which subspaces retain nondegeneracy of a bilinear form?

    Suppose I have a nondegenerate alternating bilinear form <,> on a vector space V. Under what conditions would a subspace U of V retain nondegeneracy? That is, if u ∈ U and u ≠ 0, then could I find a w ∈ U such that <u,w> ≠ 0? So for example, it's clear that no one-dimensional subspace W of V...
  6. C

    Proving matrices are subspaces

    Hi, I was wondering if someone could check my work for this linear algebra problem. I have attached the problem statement in the file "problem" and my work in the file "work." I would type out my work on here, but I couldn't figure out how to put matrices in a post so I just took a pic of my...
  7. T

    Linear Algebra subspaces and spans

    Homework Statement Let E = {“ax+by+cz = d” | a; b; c; d ∈ R} be the set of linear equations with real coefficients in the variables x, y and z. Equip E with the usual operations on equations that you learned in high school. addition of equations, denoted below by “⊕” and multiplication by...
  8. 1

    Subspaces of Functions- Linear Algebra

    Homework Statement Which of the following are subspaces of F[R] = {f |f:R-->R}? a) U = {f e F[R]|f(-1)f(1)=0 b) V = " |f(1)+f(2)=0 c) S = " |f(x)=f(-x) d) T = " |f(1)<= 0 Homework Equations The Attempt at a Solution I got S and V or c) and b), is that correct? I...
  9. D

    Fundamental subspaces of A?

    When studying linear algebra when encountering a system Ax=b, I always read of the fundamental subspaces of A: N (the null space, all solutions x of Ax=0), the column or domain space of A: (the space spanned by the columns of A, or in other words, all possible b for Ax=b), the row space (the...
  10. ElijahRockers

    Determining Polynomial Subspaces in P4

    Homework Statement Determine whether the following are subspaces of P4: a) The set of polynomials in P4 of even degree b) The set of all polynomials of degree 3 c) The set of all polynomials p(x) in P4 such that p(0) = 0 d) The set of all polynomials in P4 having at least one real root The...
  11. M

    Solving: Vector Subspaces Question in R3

    Could someone please help me with the following question with a guided step by step answer: Show that T = (x, y, z) : -1 ≤ x + y + z ≤ 1 is not a vector subspace of R3 Thanks!
  12. C

    Skew-symmetric matrices and subspaces

    Homework Statement Let W1 be the set of all nxn skew-symmetric matrices with entries from a field F. Assume F is not characteristic 2 and let W2 be a subspace of Mnxn(F) consisting of all nxn symmetric matrices. Prove the direct sum of W1 and W2 is Mnxn(F). Homework Equations The...
  13. A

    Exploring Orthogonality: Combining Bases from Subspaces in Linear Algebra

    Hi: I have a problem about combine bases from subspaces. This is part of orthogonality. The examples as following: For A=##\begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix}## split x= ##\begin{bmatrix} 4 \\ 3 \end{bmatrix}## into ##x_r##+##x_n##=##\begin{bmatrix} 2 \\ 4...
  14. J

    How Do You Find the Basis and Equation for Specific Subspaces in R^3?

    Homework Statement -Problem number 1 Given the set {u ,v} , where u=(1,2,1) and v=(0,-1,3) in R^3 find an equation for the space generated by this set. -Problem number 2 The subspace S is defined as S= {(x,y,z) : x + 2y - z =0} find a set B={u,v} in R^3 such that each...
  15. K

    Lagrangian subspaces of symplectic vector spaces

    Homework Statement If (V,\omega) is a symplectic vector space and Y is a linear subspace with \dim Y = \frac12 \dim V show that Y is Lagrangian; that is, show that Y = Y^\omega where Y^\omega is the symplectic complement. The Attempt at a Solution This is driving me crazy since I...
  16. A

    About vector space and subspaces

    Hi: I am a newbie to linear algebra; I have a problem about vector space and subspaces. How to distinguish these two subject. what I know from books is subspace is going through zero, but I still can not figure out what is the difference between vector space and subspaces, thanks.
  17. O

    Linear algebra proof subspaces

    Homework Statement Let A be a fixed 2x2 matrix. Prove that the set W = {X : XA = AX} is a subspace of M2,2. Homework Equations Theorem: Test for a subspace If W is a nonempty subset of a vector space V, then W is a subspace of V if and only if the following closure conditions hold. 1...
  18. M

    Concerning Subspaces of Infinite Dimensional Vector Spaces

    I have a question concerning subspaces of infinite dimensional vector spaces. Specifically given any infinite dimensional vector space V, how might one construct an infinite decreasing chain of subspaces? That is: V=V0\supseteqV1\supseteq... , where each Vi is properly contained in Vi-1...
  19. S

    Riemannian Submersions: Understanding the Definitions and Well-Definedness

    Hi! I have the following statements in a script on Riemannian submersions: (\pi is the submersion \mathbb S^{2n+1} \rightarrow \mathbb{CP}^n or \mathbb S^{4n+3} \rightarrow \mathbb{HP}^n respectively.) Regarding a) it is then said: "Let w \in T\mathbb{CP}^n, \lambda \in \mathbb C. Let...
  20. J

    Question on subspaces and spans of vector spaces

    Hi, I have read my notes and understand the theory, but I am having trouble understanding the following questions which are already solved (I am giving the answers as well). The first question says: Let U_{1} and U_{2} be subspaces of a vector space V. Give an example (say in V=\Re^{2}) to...
  21. C

    Linear algebra: Vector subspaces

    Homework Statement Is the subset of P= {a0 + a1x + a2x2 + ... + anxn} formed only by the polynomials that satisfy the condition: a1a3≤0 a vector subspace? Homework Equations I already proved the subset is not closed under addition so I know it's not a vector subspace, however, the...
  22. S

    Are these subspaces a vector space?

    Homework Statement Consider the ordinary vectors in three dimensions (ax, ay, az) with complex components. a) Does the subset of all vectors with az = 0 constitute a vector space? If so, what is its dimension; if not; why not? b) What about the subset of all vectors whose z component is 1...
  23. S

    How check for subspaces in Linear Algebra?

    Homework Statement consider R^4. Let V be the set of vectors in the form ( 2x+3y, x, 0 , -x+2x) is this a subspace of R^4 and why? find a basis if it's possible Homework Equations The Attempt at a Solution I know that the set must work under scalar multiplication and vector...
  24. T

    Linear Algebra: intersection of subspaces

    Homework Statement I'm working on a problem that involves looking at the dimension of the intersection of two subspaces of a vector space. Homework Equations M \subset V N \subset V dim(M \cap N) [\vec{v}]_{B_M} is the coordinate representation of a vector v with respect to the...
  25. Fredrik

    Linear or conjugate operators and automorphisms on the lattice of subspaces

    I'm reading about symmetries in QM in "Geometry of quantum theory" by Varadarajan. In one of the proofs, he refers to theorem 2.1, which is stated without proof. He says that the theorem is proved in "Linear algebra and projective geometry" by Baer. That isn't very helpful, since he doesn't even...
  26. E

    Linear algebra help: Subspaces

    Homework Statement Prove that C(AB) is a subset of C(A) for matrices A,B, where C denotes column space. Homework Equations C(AB) = {b \in \mathbbcode{R}^m: Ax=b is consistent} The Attempt at a Solution I don't really know where to start.
  27. T

    Subspace Determination using (x,y,z) = a(1,0,1) + b(0,1,0) + c(0,1,1) in R^3

    Homework Statement The Attempt at a Solution Let (x,y,z) be arbitrary. We write, (x,y,z) = a(1,0,1) + b(0,1,0) + c(0,1,1) for a,b,c \in R . From this, (x,y,z) = (a,0,a) + (0,b,0) + (0,c,c) = (a,b+c,a+c). However, (a,b+c,a+c) can generate all of R^3 for appropriately chosen a,b,c...
  28. T

    Intersection and Addition of Subspaces

    Homework Statement http://img824.imageshack.us/img824/3849/screenshot20120122at124.png The Attempt at a Solution Let S = \left\{ S_1,...,S_n \right\} . If L(S) = V, then T = \left\{ 0 \right\} and we are done because S + T = V. Suppose that L(S) ≠ V. Let B_1 \in T such that B_1 \notin...
  29. T

    Intersections of Subspaces and Addition of Subspaces

    Homework Statement http://img824.imageshack.us/img824/3849/screenshot20120122at124.png The Attempt at a Solution Let S = \left\{ S_1,...,S_n \right\} . If L(S) = V, then T = \left\{ 0 \right\} and we are done because S + T = V. Suppose that L(S) ≠ V. Let B_1 \in T such that B_1 \notin...
  30. T

    What is the proof that x, x^2, x^3 form a basis of V?

    Homework Statement http://img856.imageshack.us/img856/5586/screenshot20120121at328.png The Attempt at a Solution I propose the vectors x,x^2,x^3 form a basis of V. To test for linear independence, let 0 = a_1 x + a_2 x^2 + a_3 x^3, where a \in R. A polynomial is 0 iff all of its...
  31. T

    Identifying Subspaces and Subspace Addition

    Homework Statement http://img21.imageshack.us/img21/4580/screenshot20120117at218.png The Attempt at a Solutiona) Suppose we have two arbitrary vectors of E, call them X,Y. Let X = (2x,x) where x is in R and let Y = (2y,y) where y is in R. If we add X and Y we have (2x,x) + (2y,y) =...
  32. G

    Defining the Direct Sum of Subspaces: Can It Be Defined When k=1?

    Homework Statement Hi, everybody! I'd like to ask you about the direct sum of subspaces... I refer to two linear algebra books; 1)Friedberg's book, 2)Hoffman's book. First of all, I write two definitions of direct sum of subspaces... in the book 1), Def.1). Let...
  33. T

    Vector Subspaces Homework: The Attempt at a Solution

    Homework Statement http://img857.imageshack.us/img857/548/screenshot20120112at853.png The Attempt at a SolutionI reasoned that if U is a vector subspace, then the zero vector must certainly be an element of U. That is, (0,0,0) \in U. If this is true, then we can write for x_1 + x_2 + x_3...
  34. U

    Orthogonal Complements of complex and continuous function subspaces

    Homework Statement I'm having a tough time figuring out just how to get the orthogonal complement of a space. The provlem gives two separate spaces: 1) span{(1,0,i,1),(0,1,1,-i)}, 2) All constant functions in V over the interval [a,b] Homework Equations I know that for a subspace W of an...
  35. D

    Dimension of intersection of subspaces proof

    Homework Statement V is a vector space with dimension n, U and W are two subspaces with dimension k and l. prove that if k+l > n then U \cap W has dimension > 0 Homework Equations Grassmann's formula dim(U+W) = dim(U) + dim(W) - dim(U \cap W)The Attempt at a Solution Suppose k+l >n. Suppose...
  36. C

    Union and addition of two subspaces a subspace?

    Homework Statement This problem is broken into 5 parts: (1) Let E={(2a,a)|a∈ℝ}. Is E a subspace of R2? (2) Let B={(b,b)|b∈ℝ}. Is B a subspace of R2? (3) What is E\capB? (4) Is E\cupB a subspace of R2? (5) What is E+B Homework Equations E={(2a,a)|a∈ℝ} B={(b,b)|b∈ℝ} The Attempt...
  37. C

    Determining if vectors in R3 are linear subspaces

    Homework Statement U={(x1,x2,x3)\inℝ3 | x1+x2=0} Is this a linear subspace of ℝ3?Homework Equations x1+x2=0The Attempt at a Solution I know that in order to be a linear subspace, it must be closed under addition and scalar multiplication. I'm just not really sure how to incorporate the x1+x2=0...
  38. V

    Finding orthonormal basis for the intersection of the subspaces

    Homework Statement Homework Equations can someone help me to solve this problem? The Attempt at a Solution I couldn't even approach
  39. K

    Why is it confusing to determine whether sets form subspaces in ℝ2?

    Homework Statement Determine whether the following sets form subspaces of ℝ2: (a) {(x1, x2)T | x1 + x2 = 0} (b) {(x1, x2)T | x1 * x2 = 0}Homework Equations The Attempt at a Solution I know that a is a subspace and b is not, but I would like to know why. For part A, I let x=[c, -c]T ∂[c,-c]=...
  40. S

    Finding the Projection onto Subspaces

    Homework Statement See attachment The Attempt at a Solution How should I approach these questions? By using the projection formula?
  41. S

    Bases, Subspaces, Orthogonal Complements and More to Come

    Homework Statement Show that the set W consisting of all vectors in R4 that are orthogonal to both X and Y is a subspace of R4. Here X and Y are vectors such that X = (1001) and Y = (1010). Part b) Find a basis for W. The Attempt at a Solution So I know to satisfy being a...
  42. G

    Polynomial Subspace Dimension & Basis Calculation

    Homework Statement Let M be a subspace of the vector space \mathbb{R}_2[t] generated by p_1(T)=t^2+t+1 and p_2(T)=1-t^2, and N be a subspace generated by q_1(T)=t^2+2t+3 and q_2(T)=t^2-t+1. Show the dimension of the following subspaces: M+N, M \cap N, and give a basis for each...
  43. A

    Understanding Cosets and Subspaces in Linear Algebra

    Hi, I have just begin with Linear Algebra. I came across cosets and I don't understand what is the difference between cosets and subspaces? thanx in advance.
  44. T

    Finding intersection of vector subspaces

    Homework Statement What are the intersections of the following pairs of subspaces? (a) The x-y plane and the y-z plane in R'. (b) The line through (1, 1, 1) and the plane through (1,0, 0) and (0, 1, 1). (c) The zero vector and the whole space R'. (d) The plane S perpendicular to (1, 1...
  45. N

    Vector Subspaces: Understanding Closure Properties

    Hello :) I've been doing a lot of work on subspaces but have come across this question and need a bit of help! Homework Statement W = {(x, y) \in R^{2} | x^{2} + y^{2} = 0} Homework Equations 1. 0 ∈ W 2. ∀ u,v ∈ W; u+v ∈ W 3. ∀ c ∈ R and u ∈ W; cu ∈ W The Attempt at a...
  46. B

    Linear Algebra Subspaces Basis

    Homework Statement a) If U and W are subspaces of R^3, show that it is possible to find a basis B for R^3 such that one subset of B is a basis for U and another subset of B (possibly overlapping) is a basis for W. b) If U and W are subspaces of a finite-dimensional vector space V, show...
  47. F

    Proving that the interesection of subspaces is a subspace

    Homework Statement Let U1; U2 be subspaces of the vector space V . Prove that their intersection U1 \ U2 is also a subspace of V Homework Equations I see how any equations could be used here The Attempt at a Solution Well intuitively this seems obvious from the get go. If U1 and...
  48. A

    Vector Spaces & Subspaces: Proving Addition Closure

    Homework Statement [PLAIN]http://i26.lulzimg.com/274748.jpg Homework Equations ?? The Attempt at a Solution i don't even know how to start. lol.
  49. S

    Proof involving vector subspaces

    Let W1 and W2 be subspaces of a vector space V. Prove that W1\bigcupW2 is a subspace of V if and only if W1\subseteqW2 or W2\subseteqW1Well so far, I have proven half of the statement (starting with the latter conditions). Right now I'm struggling to show that the final conditions follow from...
  50. W

    Proving T is a Subspace of M23: Linear Algebra Problem

    Hi there. I started learning about subspaces in linear algebra and I came across a question which I'm unsure how to solve. I understand that there are 'rules' which need to be passed in order for something to be a subspace, but I have no idea how to start with this problem: Consider the set...
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