What is Subspace: Definition and 572 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. T

    Proving Uniqueness in Subspace Addition

    Homework Statement http://img854.imageshack.us/img854/5683/screenshot20120116at401.png The Attempt at a SolutionSo we have that A + B is a vector in S + T, where A is an element of S and B is an element of T. Suppose there is another vector A' + B' also in S + T, where A' is an element of S...
  2. C

    Krylov subspace iterative method for large sparse matrix

    Hey! I have a project for a Circuits Simulation class which is basically the programming of a small spice software. So the program must be able to find dc solutions and do transient analysis. This means it can solve Ax = b for a large (think 10^6 X 10^6) sparse non-symmetric matrix A which is...
  3. C

    Subspace of P3, linearly independence?

    Homework Statement Let U be the subspace of P3(ℝ) spanned by E={x^3,x^3-x^2,x^3+x^2,x^3-1} find a linearly independent subset F of E spanning U. Homework Equations E={x^3,x^3-x^2,x^3+x^2,x^3-1} The Attempt at a Solution a(x^3)+b(x^3-x^2)+c(x^3+x^2)+d(x^3-1)=0x^3+0x^2+0x+0...
  4. 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...
  5. C

    Figuring out if subset of R2 is a subspace

    Homework Statement Let E be the subset of R2 defined by E={(x,y)|x≥0,y\inℝ}. Is E a subspace of R2? Homework Equations E={(x,y)|x≥0,y\inℝ} The Attempt at a Solution I honestly have no idea where to start. Please help ! I'm not asking for the answer per se, just a starting point
  6. C

    Show that two sets of vectors span the same subspace

    Homework Statement Show that the two sets of vectors {A=(1,1,0), B=(0,0,1)} and {C=(1,1,1), D=(-1,-1,1)} span the same subspace of R3. Homework Equations {A=(1,1,0), B=(0,0,1)} {C=(1,1,1), D=(-1,-1,1)} The Attempt at a Solution aA+bB=(a,a,0)+(0,0,b)=(a,a,b)...
  7. C

    Find vectors that span a subspace

    Homework Statement Find a pair of vectors that span the subspace x+y-2z=0 of R3.Homework Equations x+y-2z=0The Attempt at a Solution I just guessed some numbers since its such a simple equation and came up with (1,-1,0) and (2,0,1). I was just wondering what the standard method is to figure...
  8. S

    A question about subspace topolgy

    Let (X, τ) be a topolgical space. Let f: τ→R be a map that assigns real values to the elements of τ. Let (A,τ_A) be subspace of (X,τ). Let g:τ_A→R be another map that assigns real values to the element of subspace topology. My question is how the function g is related with function f given that...
  9. H

    [Linear Algebra] Subspace Proof

    Homework Statement I have a matrix and need to show that it is a subspace of ℝn using the eigenspace identity of: Ax = λx, where x is the eigenvector. Homework Equations The Attempt at a Solution I know that for a subspace, you need to show that it holds under addition, scalar...
  10. B

    Isomorphism: subspace to subspace?

    Homework Statement We're looking at a mapping from P2 (polynomials of degree two or less) to M2(R) (the set of 2x2 real matrices). The nuance here is that the transformation into the matricies is such that its basis consists of only three independent matrices, making its dimension 3. This...
  11. D

    Can F4 Be a Subspace If b Is Nonzero?

    Homework Statement Show that if b \in F then {(x_1, x_2, x_3, x_4)\in F_4 : x_3 = 5x_4 + b} is a subspace of F_4 if and only if b = 0. F is defined as either ℝ or ℂ. The Attempt at a Solution I'm still trying to get the hang of these "proofs." Let c \in F to check if this is...
  12. S

    Find Basis for Subspace of R^4 Spanned by Vectors

    Hello. First, I'd like to apologize because I don't know where to go ask for homework on linear algebra on the forums so if anyone could please let me know, that would be appreciated. Here's the question: Find a basis for the subspace of R^4 spanned by the given vectors Here's the answer...
  13. T

    Finding a subspace (possibly intersection of subspace?)

    Homework Statement Let A be the following 2x2 matrix: s 2s 0 t Find a subspace B of M2x2 where M2x2 = A (+) B Homework Equations A ∩ B = {0} if u and v are in M2x2, then u + v is in M2x2 if u is in M2x2, then cu is in M2x2 The Attempt at a Solution Let B be the...
  14. R

    Note that you can use the same c1, c2, and c3 for both parts.

    Homework Statement 1) Is S a subspace of R^n? 1.1) Given n=4 and a vector is in S if it is in the span of e1, e2 or in the span of e3, e4 where e1...e4 is the canonical basis of R^4 1.2) Given n=3 and S is a sphere of radius 1. 2) Let S be a subspace of R^10 with basis v1; v2; v3. Show that...
  15. H

    Adjoint of linear Operator and T-invariant subspace

    Homework Statement Let T be a linear operator on an inner product space V and W be a T-invariant subspace of V. If W is both T and T* invariant, prove that (T_{W})* = (T*)_{W}. Note that T_{W} denotes the restriction of T to W Homework Equations \forallx\inW, T_{W}(x) = T(x)...
  16. N

    6 orthonormal subspace question

    6)there is normal T in unitarian final space. v\neq0,v\in V prove that if \{sp(v)\}^{\perp} is T variant then v is eigenvector of T ? hint:prove that T*(v) is orthogonal to \{sp(v)\}^{\perp} what i have done: suppose u\in\{sp(v)\}^{\perp} we take the definition of T* (Tu,v)=(u,T*v)...
  17. B

    Is A Closed Under Scalar Multiplication in W?

    Is the subset A a subspace of W W=\left \{ \begin{bmatrix} 1 &1 \\ a_{21}& a_{22} \\ a_{31}& a_{32} \end{bmatrix} :a_{ij} \in \mathbb{C}\right \} Let A=\begin{bmatrix} 1 &1 \\ a_{21}& a_{22}\\ a_{31}& a_{32} \end{bmatrix} A \in W Then 2A \in W since...
  18. N

    Find a basis of U, the subspace of P3

    Homework Statement Find a basis of U, the subspace of P3 U = {p(x) in P3 | p(7) = 0, p(5) = 0}Homework Equations The Attempt at a Solution ax3+bx2+cx+d p(7)=343a+49b+7c+d=0 p(5)=125a+25b+5c+d=0 d=-343a-49b-7c d=-125a-25b-5c ax3+bx2+cx+{(d+d)/2} -->{(d+d)/2}=2d/2=d...
  19. B

    Vector Subspace: Is W a Subspace of V?

    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 entries is a subspace of V Homework Equations I got this far x=(x_n), y=(y_n) be elements of W, then there...
  20. K

    Showing two sets are not homeomorphic in subspace topology.

    Homework Statement The true problem is too complicated to present here, but hopefully somebody can give me a hand with this simplified version. Consider the set H = \{ (x,y) \in \mathbb R^2 : y \geq 0 \} . Denote by \partial H = \{ (x,0) \}. Let U and V be open sets (relative to H) such that...
  21. 3

    Determining Subspace of Singular 2x2 Matrices in R^2

    Homework Statement Determine if the set of all singular 2 x 2 matrices are a subspace of R^{2} Homework Equations If a, b, c, and d are the entries of a 2 x 2 matrix, then their determinant, ad - bc = 0 if the matrix is singular. The Attempt at a Solution I have been doing other...
  22. D

    Difference between a vector subspace and subset?

    This is quite confusing to me. I know a vector subspace is a vector space within another vector space and is closed under the operations of the vector space it lies in, but how exactly does it differ from vector subsets? Anyone care to explain or clarify this? My textbook is completely terrible...
  23. K

    Prove that Wm is a subspace of R2?

    I need help with this problem that I don't know how to solve. Homework Statement For each positive integer m, it's defined a subset of R2 as Wm={(mx,x)|x in R} (a) Prove that each Wm is a subspace of R2. (b) ¿Is the union of all Wm a subspace of R2?. Prove it. Homework Equations None. The...
  24. B

    Is the following set a subspace?

    Homework Statement x belongs to the vector space R^6. Is (x1-x2)^4 + x3^6 = 0 a subspace? Homework Equations Since we already know x is a vector space we only need to check: 1. The existence of the zero vector 2. Closure under vector addition 3. Closure under scalar addition The Attempt...
  25. N

    Proving if a subset is a subspace

    Homework Statement Is the following a subspace of R^{n} for some n? W = {(x, y, z) \in R^{3} | 2x - y = 3z + x = 0} Homework Equations A subspace of R^{n} is a subset W of R^{n} such that; 1. 0 \in W 2. \forall u, v \in W; u + v \in W 3. \forall c \in R and u \in W; cu \in W...
  26. F

    Proving that a subspace must have a specific number of elements

    Homework Statement Let V = (F2)^3, the set of triples (x; y ; z) of numbers in F2, the fi eld with two elements. V is a vector space over F2. Prove that any subspace of V must have either 1, 2, 4, or 8 elements. Homework Equations F2 = {0,1} The Attempt at a Solution The only...
  27. 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...
  28. N

    Subspace Question: Why Not Closed Under Addition?

    Homework Statement [PLAIN]http://img683.imageshack.us/img683/4530/unledkw.jpg can someone please explain why it is not closed under addition? My textbook did not explain very well, but I understand this can be zero vector and it is closed under scalar multiplication. thanks!Homework Equations...
  29. DryRun

    Is a set of orthogonal basis vectors for a subspace unique?

    Homework Statement Is a set of orthogonal basis vectors for a subspace unique? The attempt at a solution I don't know what this means. Can someone please explain? I managed to find the orthogonal basis vectors and afterwards determining the orthonormal basis vectors, but I'm not sure what the...
  30. D

    Proving Subspace of Mnn: AB=BA for Fixed nxn Matrix B

    Homework Statement Prove that the set of all n x n matrices A such that AB = BA for a fixed n x n matrix B, is a subspace of Mnn. Homework Equations u + v is in the same vector space as u and v. ku is in the same vector space as u, where k is any real number. The Attempt at a...
  31. M

    State space and its subspace : finding a basis

    Hello. I really need help with this one: Homework Statement I have a 3 dimensional state space H and its subspace H1 which is spanned with |Psi> = a x1 + b x2 + c x3 and |Psi'> = d x1 + e x2 + f x3 Those two "rays" are linearly independent and x1, x2, and x3 is an...
  32. W

    Proving the Subspace Property of U + W

    Homework Statement Let U and W be subspaces of a vector space V Show that the set U + W = {v ∈ V : v = u + w, where u ∈ U and w ∈ W} is a subspace of V Homework Equations The Attempt at a Solution I understand from this that u and w are both vectors in a vector space V and that u+w...
  33. A

    Subspace of a Vector Space over Complex Numbers Proof.

    Homework Statement Let V = C (complex numbers). Prove that the only C-subspaces of V are V itself and {0}. Homework Equations The Attempt at a Solution Well this problem has me confused since I have clearly found a complex subspace for example all the complex numbers of the form {a+ib ...
  34. N

    Is S a Valid Subspace of R^4?

    Homework Statement Show that the set: S = {x \in R^{4}| x = \lambda(2,0,1,-1)^{T} for some \lambda \in R is a subspace of R^{4} The Attempt at a Solution For the subspace theorem to hold, 3 conditions must be met: 1) The zero vector must exist 2) Closed under addition 3)...
  35. D

    Proving Subspace: U + W in Vector Space V

    Homework Statement Let U and W be subspaces of a vector space V. Show that the set U + W = {v (element symbol) V : v = u + w, where u (element symbol) U and w (element symbol) W} Is a subspace of V. Homework Equations - The Attempt at a Solution I really don't know where...
  36. D

    T invariant subspace (intro lin alg class undergrad)

    Homework Statement V=Matrix (2x2), T(A) = (0 1 ) A , and W = {A\epsilon V: A^{}t = A (1 0) Homework Equations So T(A) transformation, multiplies a 2x2 matix with entries 0 1 1 0 by A with A on the right side The Attempt at a Solution I...
  37. S

    Is W a Subspace of V? Exploring 3x3 Matrices

    Homework Statement Let V be the spcae of all 3x3 matrices with real entries. Is W, the set of all 3x3 lower triangular matrices, a subspace of V? Why or why not? Homework Equations The Attempt at a Solution I just think that all 3x3 lower triangular matrices are included in...
  38. J

    Find an orthogonal basis for the subspace of

    Homework Statement ... R4 consisting of all vectors of the form [a+b a c b+c] Homework Equations Gram-Schmidt process, perhaps? The Attempt at a Solution Not sure how to approach this one. Helpful hint?
  39. J

    Is W a Subspace of R^3 and What is Its Geometry?

    I'm stuck on a problem which asks: Determine whether W is a subspace of R^3. If W is a subspace, then give geometric description of W. The problem is W={x:x3=2x1-x2} and x=[x1, x2, x3] I tried solving it but I'm having a hard time understanding the properties of R^n and using them. I guess I'm...
  40. M

    Does the set of solutions for Ax=b form a subspace?

    I have a non-homogeneous Ax=b (with b non-zero) and i want to know if the set of all the solution vectors, x, forms a subspace. I know that every solution can be written as x = xparticular + xhomogeneous i.e as the sum of a particular solution and a homogeneous solution, but I'm not sure if...
  41. M

    What is a Closed Linear Subspace?

    Hi. I'm trying to find a good definition of a closed linear subspace (as opposed to any other linear subspace), and I can't find anything concise and comprehensible. Any help will be much appreciated. P.S. I'm not great at analysis, so please try to keep it simple.
  42. T

    Linear Algebra: Subspace Sum: What is U + U?

    Homework Statement Suppose U is a subspace of V. What is U + U? Homework Equations There are two definitions of a subspace sum that I know of (the first is the definition given in my book): (1) U_1 + U_2 = \{ u_1 + u_2 : u_1 \in U_1, u_2 \in U_2 \} (2) U_1 + U_2 = \text{ span} ( U_1...
  43. A

    Why an Invariant Subspace Has an Eigenvector

    I am following a proof in the text "Algebras of Linear Transformations" and having problem justifying this line: ... M is an invariant subspace so it has an eigenvector. Why should an invariant subspace have an eigenvector? Thank you I have a feeling this is a very simple result, if so I am sorry
  44. E

    Definitions of vector space and subspace

    I am using Axler's Linear Algebra Done Right as a text for independent study of linear algebra. Axler basically defined a vector space to be a set which has defined operations of addition and multiplication (and which comports with certain algebraic properties) and that contains an additive...
  45. A

    Is W1 a Vector Subspace with Only the Zero Vector?

    Is the following a vector subspace W1 {(x1,x2,x3): x1=x2=x3=0} I usually begin my attempt by finding two members of the set then check which axioms are valid.However I can only think of 1: (0,0,0) Any help would be great thank you
  46. R

    Show the set S is a subspace of Real Numbers^3

    Homework Statement Show that set S = {(x , y, z ) | x + 2y − z = 0} is a subspace of Real Numbers^3. Homework Equations A subspace needs to be closed under addition and scalar multiplication The Attempt at a Solution S = { (x, y, x+2y) | x, y are elements of Real Numbers }...
  47. Y

    A subspace has finite codimension n iff it has a complementary subspace of dim nu

    Homework Statement A subspace N of a vector space V has finite codimension n if the quotient space V/N is finite-dimensional with dimension n. Show that a subspace N has finite codimension n iff N has a complementary subspace M of dimension n. Do not assume V to be finite-dimensional. 2...
  48. S

    Is there a symbol for indicating one vector space is a subspace of another?

    Hi all, I was just wondering, is there is a particular symbol to say V is a subspace of W? I suppose V\subsetW works if I describe each (sub)space in set notation first, but I was wondering what I could use if I don't state W or V as a particular set? Thanks
  49. Y

    (n-1)-dimensional subspace is the null space of a linear functional

    Given that N is an (n-1)-dimensional subspace of an n-dimensional vector space V, show that N is the null space of a linear functional. My thoughts: suppose \alpha_i(1\leq i \leq n-1) is the basis of N, the linear functional in question has to satisfy f(\alpha_i)=0. Am I correct? Thanks
  50. H

    How to show something is a subspace

    How would you show that {(x,y,z) € R^3 :√11x - √13z=0} is a subspace of R^3? I know you have to make sure it fits the definition of a subspace, i.e prove u+v € W and alpha(v) € W but I am not sure how you would do this using √11x - √13z=0 ?
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