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. T

    Find a Basis for the Subspace of R4

    Homework Statement Find a basis for the subspace of R4 spanned by S. Homework Equations S: {(2,9,-2,53), (-3,2,3,-2), (8,-3,-8,17), (0,-3,0,15)} I've attempted this using a matrix and row reducing it. I'm just not sure if there's another simpler way, as I keep on getting incorrect...
  2. H

    Independent Subspace: Proving (or Disproving) Linear Independence

    Suppose B = {b1,...,bn} and C={c1,...,cn} both are basis set for space V. D = {d1,...,dn} is basis for space T. If B and D is linearly independent, is C and D always independent too? How can we prove (disprove) it?
  3. A

    Complete subspace of function space

    Homework Statement (This is for a functional analysis course.) Let C(X,Y) be the space of continuous functions X \to Y. Let \mathcal{C}_0 = \{\varphi \in C([0,\infty),\mathbb{R}) : lim_{x\to \infty}\varphi(x) = 0 \}. Let \mathcal{C}_0^1 = \{\varphi \in \mathcal{C}_0 : \varphi' \in...
  4. D

    Have I proved W is a subspace of R3?

    Homework Statement Let W = { (x, y, z) | x - 2y + z = 0 } Is W a subspace of R3? Homework Equations The Attempt at a Solution Using another post here I tried the following to show W is closed under addition: 1. Let u = (x, y, z) and v = (i, j , k). u and v are both in W...
  5. S

    Show a subspace and find a spanning set

    Homework Statement Let W be the subset of all P3 defined by W={p(x) in P3: p(1)=p(-1) and p(2)=p(-2)} Show that W is a subspace of P3, and find a spanning set for W. Homework Equations The Attempt at a Solution This is homework we can correct. I've attached my work. The...
  6. T

    Upper triangular matrix as subspace

    Homework Statement Which of the following subsets of Mn,n are subspaces of Mn,n with the standard operations: The set of all n x n upper triangular matrices Homework Equations 10 axioms of vector spaceThe Attempt at a Solution The set of all n x n upper triangular matrices is not closed...
  7. R

    Subspace of Polynomials: \vec{}p(t)=a+t^2, where a is real

    Homework Statement Determine if all polynomials of the form \vec{}p(t)=a+t^2, where a is in real, are a subspace of \Re_{}n.The Attempt at a Solution the correct answer says that p(t) is not a subspace since the zero vector is not in the set. im trying to work this out and got P(0)=a+0=a is...
  8. I

    Ortogonal subspace proof - Leon 5.2

    Homework Statement Let S be a subspace of R3 spanned by the vectors x = (x1, x2, x3)T and y = (y1, y2, y3)T Let A = (x1 x2 x3 ) ( y1 y2 y3) Show that S\bot = N(A). Homework Equations The Attempt at a Solution Any hints?
  9. F

    Solving Vector Subspace Questions: A & B in V

    Hey guys this is the question . Let A and B be vector subspaces of a vector space V . The intersection of A and B, A ∩ B, is the set {x ∈ V | x ∈ A and x ∈ B}. The union of A and B, A ∪ B, is the set {x ∈ V | x ∈ A or x ∈ B}. a) Determine whether or not A ∩ B is a vector subspace of V ...
  10. D

    True or False: P-Dimensional Subspace and Basis for R^n

    Homework Statement If H is a p-dimensional subsapce for R^n and {v1,...vp} is a spanning set of H, then {v1,...vp} is automatically a basis for H. True or False Homework Equations I am unsure of my answer. The Attempt at a Solution I am under the impression that this is...
  11. H

    Subspace math techniques

    1) Let x0 be a fixed vector in a vector space V. Show that the set W consisting of all scalar multiples cx0 of x0 is a subspace of V. What techniques should I use to prove this? 2a) Show that a line lo through the origin of R^n is a subspace of R^n. 2b) show that a line l in R^n not...
  12. M

    Proving that W1 \cap W2 is a Subspace of V

    Prove that if W1 and W2 are subspaces of the vector space V, then W1 \cap W2 is also a subspace of V. Attempt at solution: I really don't even know where to start on this because I am confused about how to prove in general that something is a subspace. Also, I don't know how to find what W1...
  13. H

    Subspace for Linear Alegbra

    1.) The set W of all 2x3 matrices of the form a b c a 0 0 where c = a + b, is a subspace of M23 (Matrics 23). Show that every vector in W is a linear combination of W1 = 1 0 1 1 0 0 W2 = 0 1 1 0 0 0 Do I have to combine both...
  14. D

    Is {a0 + a1x + a2x^2 + a3x^3 | a0a3 - a1a2 = 0} a subspace of P3?

    Homework Statement Is {a0 + a1x + a2x^2 + a3x^3 | a0a3 - a1a2 = 0} a subspace of P3? Why or why not? *The digits should be in subscript. How would I go about answering this?
  15. Z

    Find bases for the following subspace of F^5

    Homework Statement Find bases for the following subspaces of F^5: W1 = {(a1, a2, a3, a4, a5) E F^5 : a1 - a3 - a4 = 0} and W2 = {(a1, a2, a3, a4, a5) E F^5: a2 = a3 = a4 and a1 + a5 = 0} 2. The attempt at a solution Well, I understand a basis is the maximum amount of vectors...
  16. S

    Basis of subspace (and combinations of them)

    Homework Statement We are given the following subspaces U := {x E R3: x1 + 2*x2 - x3 = 0} and V := {x E R3: x1 - 2*x2 - 2*x3 = 0} And we need to find a basis for (i) U (ii) V (iii) U n V (not an "n" but a symbol that looks like an upside-down U) (iv) span(U u V) (not a "u" but a symbol that...
  17. T

    Invariant subspace and linear transformation

    Homework Statement Let U be a subspace of V. Suppose that U is a invariant subspace of V for every linear transformation from V to V. Show that U=V. Homework Equations no The Attempt at a Solution Assume U is not trivial: Now we only need to show that U = V. Let dimV = n: We can...
  18. Demon117

    Vector subspace F is closed in E

    Let E be the vector space of bounded functions f:N --> R, with the norm(g) = sup|f|. Assume without proof that the norm holds, so that the function d(f,g)=norm(f - g) is a metric. Prove that the vector subspace F={f in F | f(n) -->0 as n --> infinity} is closed in E. Here is what I have...
  19. A

    Decide whether or not U is a subspace of V

    Homework Statement For each of the following subsets U of the vector space V I have to decide whether or not U is a subspace of V . In each case when U is a subspace, I also must find a basis for U and state dim U: (i) V = R^4; U = {x = (x1; x2; x3; x4) : 3x1 - x2 -2x3 + x4 = 0}: (ii) V =...
  20. C

    Determining a subspace of polynomials with degree 3

    Homework Statement Determine which of the following are subspaces of P3: a) all polynomials a0+a1x+a2x^2+a3x^3 where a0=0 b) all polynomials a0+a1x+a2x^2+a3x^3 where a0+a1+a2+a3=0 c) all polynomials a0+a1x+a2x^2+a3x^3 for which a0, a1, a2, a3 are integers d) all polynomials of the form...
  21. R

    Exploring the Subspace of a Homogeneous System of Linear Equations

    Homework Statement Suppose you have points of a specific form, say (x, y, 3x + 2y). Show that this set of points is a solution to a homogeneous system of linear equations, hence a subspace. The Attempt at a Solution I'm wondering how one is able to go about this. Here's my try, but I'm not...
  22. O

    Are all 2x2 Matrices with det(A) = 0 a Subspace of M2x2?

    Homework Statement Determine whether all 2x2 matrices with det(A) = 0 are a subspace of M2x2, the set of all 2x2 matrices with the standard operations of addition and scalar multiplication.Homework Equations Must pass in order to be a subspace Closure property of addition - If w and v are...
  23. F

    Understanding the Limitations of the Projection onto a Subspace Equation

    The Projv(x) = A(ATA)-1ATx I'm puzzled why this equation doesn't reduce to Projv(x) = IIx since (ATA)-1 = A-1(AT)-1 so that should mean that A(ATA)-1AT = AA-1(AT)-1AT = II What is wrong with my reasoning? Thanks.
  24. O

    Matrix Subspace question: Does B2x2 form a subspace of M2x2?

    Homework Statement The set of all matrices A2x2 forms a vector space under the normal operations of matrix + and Scalar multiplication. Does the set B2x2 of all symmetric matrices form a subspace of M2x2? Explain. Homework Equations AT = A Closure property of addition - If w and v are...
  25. radou

    Showing a closed subspace of a Lindelöf space is Lindelöf

    Homework Statement As the title says, one needs to show that if A is a closed subspace of a Lindelöf space X, then A is itself Lindelöf. The Attempt at a Solution Let U be an open covering for the subspace A. (An open covering for a set S is a collection of open sets whose union equals...
  26. R

    Finding the basis for a subspace in vectorspace

    Homework Statement Find the basis for the subspace S of the vector space V. Specify the dimension of S. S={a a+d} where a,d are elements of R and V= M2x2 {a+d d } Homework Equations I guess I know the standard basis for M2x2 are the [(10 00) (01 00) (00 10) (00 01)]...
  27. T

    Proving Non-Equivalence of Matrices Using Trace in Linear Algebra

    Homework Statement [PLAIN]http://img152.imageshack.us/img152/3162/linal.jpg Homework Equations The Attempt at a Solution How do I do part (i) and follow the hint?
  28. C

    Dimension of subspace of even and odd polynomials

    Homework Statement I have a question which asks me to find the dimensions of the subspace of even polynomials (i.e. polynomials with even exponents) and odd polynomials. I know that dim of Pn (polynomials with n degrees) is n+1. But how do I find the dimensions of even n odd polynomials...
  29. S

    Dimension of subspace of trace of matrix

    Let V=Mn(k) be a vector space of matrices with entries in k. For a matrix M denote the trace of M by tr(M). What is the dimension of the subspace of {M\inV: tr(M)=0} I know that I am supposed to use the rank-nullity theorem. However I'm not sure exactly how to use it. I know that the trace is...
  30. S

    R3 Subspace: Proving S={(x,y,z): √3x = √2y} is a Subspace of R3

    R3 Subspace - Urgent Homework Statement Prove that S={(x,y,z):\sqrt{}3 x=\sqrt{}2 y is a subspace of R3 I'm really confuse with this and I still don't know how to proved it. Can anyone help me with this? I really a newbie in this. >< Homework Equations The Attempt at a Solution...
  31. L

    Is the Intersection of Two Subspaces Always a Subspace?

    Homework Statement R = { (a+1, b 0) | a, b are real numbers} S = { (a+b, b, c) | a, b, c are real numbers) T = R intersect S I have shown that R and S are subspaces of R^3. Now I have to determine whether T is also a subspace of R^3. The answer provided is that yes, T is also a...
  32. L

    Is S = {(a+1,b,0)|a,b are real numbers} a Subspace of R^3?

    Homework Statement Show that S = {(a+1,b,0)|a,b are real numbers} is NOT a subspace of R^3. Homework Equations The Attempt at a Solution I take a specific counter example: Let k = 0 inside real, and u = (1+1,1,0) inside S ku = 0(1+1,1,0) = (0,0,0) not inside S So, S is...
  33. G

    Basis & Dimension: Subspace of R4

    Homework Statement Find a basis and dimension to each of the following subspaces of R4: U = {(a+b,a+c,b+c,a+b+c)|a,b,c∈R} Homework Equations The Attempt at a Solution I started by making a linear system. w(a + b) + x(a + c) + y(b + c) + z(a + b + c) = 0 a(w + x + z) + b(w...
  34. E

    Subspace Proof (using addition and multiplication)

    Homework Statement Determine whether or not W is a subset of R4 W is the set of all vectors in R4 such that x1x2=x3x4 Homework Equations Two methods. u+v (addition) cu (multiplication) The Attempt at a Solution I having trouble getting the hang of subspaces. I thought I was getting close...
  35. C

    Subspace of l2/L2 that is closed/not closed.

    Homework Statement Give a nontrivial example of an infinite dimensional subspace in l2(R) that is closed. Also find an example of an infinite dimensional subspace of l2(R) that is not closed. Repeat the same two questions for L2(R). The Attempt at a Solution To my understanding, l2 is...
  36. M

    Vertical and horizontal subspace

    where in the definition of vertical subspace we understand that the notion of canonical vertical vector: a vertical vector is a vector tangent to the fiber. ?
  37. B

    Find Basis for Subspace: S with Degree ≤ 4 & f(0)=f(1)=0

    Homework Statement I need to find a basis for the following: S = {f are polynomials of degree less than or equal to 4| f(0) = f(1) = 0} 2. The attempt at a solution A general polymial is of the form: p(x) = ax^4 + bx^3 + cx^2 + dx + e Now for p(0) = p(1) = 0 I must have: e = 0 and a + b...
  38. F

    What is a Basis for a Polynomial Subspace with Specific Roots?

    Homework Statement Let P_4(\mathbb{R}) be the vector space of real polynomials of degree less than or equal to 4. Show that {{f \in P_4(\mathbb{R}):f(0)=f(1)=0}} defines a subspace of V, and find a basis for this subspace. The Attempt at a Solution Since P_4(\mathbb{R}) is...
  39. R

    Is the following subset a subspace?

    Homework Statement Determine if the following subset of Rn is a subspace: all vectors <a1, a2, ... , an>, such that a1 = 1.Homework Equations The Attempt at a Solution I'm going through the Linear Algebra: An Introductory Approach by Curtis and found this thing. I can't quite get around the...
  40. D

    Determine if subset is subspace of R3. Need Help.

    1. {[x,y,z] | x,y,z in R, z = 3x+2}. How do I determine if this subset is a subspace of R3? Am I wrong when I say this set contains the zero vector? If this is the case, then I have to use the addition and multiplication closure methods, right? Thanks
  41. K

    Orthogonal Projection onto Hilbert Subspace

    Homework Statement I have a fixed unitary matrix, say X_d \in\mathfrak U(N) and a skew Hermitian matrix H \in \mathfrak u(N) . Consider the trace-inner product [tex] \langle A,B \rangle = \text{Tr}[A^\dagger B ] [/itex] where the dagger is the Hermitian transpose. I'm trying to find the...
  42. J

    Complexification of subspace

    Let F be a subspace of a real vector space V and let G \subset V_C i.e. a subspace of its complexification. Define the real subspace of G by G_R := G \cap V. There is a symplectic form w[u,v]. The annihilator subspace F^perp of V is defined by F^perp = {v \in V : w[u,v] = 0...
  43. W

    Vertical and horizontal subspace of a vector space T_pP.

    suppose we have a principle fiber bundle P at a point p \in P we have the decomposition T_pP=V_pP + H_pP it is said that the vertical subspace V_pP is uniquely defined while H_pP is not i cannot understand this point the complement to a unique subspace is surely unique, i think. it is a...
  44. D

    Orthogonal Projection Onto a Subspace?

    Hey, I have a linear algebra exam tomorrow and am finding it hard to figure out how to calculate an orthogonal projection onto a subspace. Here is the actual question type i am stuck on: I have spent ages searching the depths of google and other such places for a solution but with no...
  45. estro

    Special subspace of M(2*3) (R)

    W=Sp\{\left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & 3 \end{array} \right), \left( \begin{array}{ccc} 1 & 0 & 1 \\ 2 & 2 & 3 \end{array} \right), \left( \begin{array}{ccc} -1 & 1 & -1 \\ -3 & -2 & -3 \end{array} \right) \} I have to find subspace T, so that M_{2*3}(R)=W\oplus T I solved...
  46. K

    Linear Algebra: Projection onto a subspace

    Homework Statement That is the question. The answer on the bottom is incorrect Homework Equations I believe that is the formula that is supposed to be used. The Attempt at a Solution All I really did was plug in the equation into the formula but there is something I am...
  47. W

    Invertible 3x3 matrices a subspace of 3x3 matrices

    Homework Statement Is the set of invertible 3x3 matrices a subspace of 3x3 matrices? Homework Equations The Attempt at a Solution I think no - the 'neutral 0 element' is not in the subset since the 3x3 0 matrix is not in the subset. Am I right? The book says it's not a subspace...
  48. W

    Set of degree 2 polynomials a subspace

    Homework Statement Which of the subsets of P2 given in exercises 1 through 5 are subspaces of P2? Find a basis for those that are subspaces. (P(t)|p(0) = 2) Homework Equations The Attempt at a Solution The solution manual says that this subset is not a subspace because it...
  49. A

    Subspace of A and a matrix formed by row operation

    The subspace formed by matrix A and A' will be same or different if A' is obtained by applying an elementary row operation on A? Please prove it.
  50. S

    Invariant Subspace: Understanding Definitions

    So I'm trying to get an idea of what an invariant subspace is and so please let me know if my understanding is correct. Given that you have some vector subspace being a collection of a particular number of vectors with the the space denoted as |\gamma>. If you have some other collection of...
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