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

    Proving a Vector Space Cannot be the Union of Two Proper Subspaces

    Prove that a vector space cannot be the union of two proper subspaces. Let V be a vector space over a field F where U and W are proper subspaces. I am not sure where to start with this proof.
  2. J

    Question about projections and subspaces

    If we want to caculate the projection of a single vector, v=(1,2) (which is an element of an R2 vector space called V) onto the subspace of V (which we call W), do we use projection of v onto W = <v,w1>w1 + <v,w2>w2 + ... <v,wn>wn However, if the individual values of v are not known (that...
  3. T

    Can't get my head into vector spaces and subspaces

    Hi, I'm new to linear algebra. I'm pretty good at doing exercises with matrices and stuff but even though I've been looking in different books, looking all over the internet I can't get into vector spaces and subspaces. It seems like the books have some very elementary and simple examples and...
  4. M

    Question relating to homogeneous system, subspaces and bases.

    Homework Statement Let A be an m x n matrix such that the homogeneous system Ax=0 has only the trivial solution. a. Does it follow that every system Ax=b is consistent? b. Does it follow that every consistent system Ax=b has a unique solution? The Attempt at a Solution So if the homogeneous...
  5. T

    Linear Algebra: Subspaces proof

    Homework Statement Suppose U and W are subspaces of V, and U \, \bigcup \, W is a subspace of V. Show that U \subseteq W. The Attempt at a Solution I have been working on this one for a bit and have not made any headway. I wish I could post anything, even a start to this one but I...
  6. T

    Is the Intersection of Subspaces Always a Subspace?

    Homework Statement Prove that the intersection of any collection of subspaces of V is a subspace of V. Homework Equations To show that a set is a subspace of a vector space, I need to show that there exists an additive identity, and that the set is closed under addition and scalar...
  7. T

    Linear Algebra: quick little question about sums of subspaces

    Homework Statement Learning about sums of subspaces and wanted to be sure that I am understanding this correctly. Say that I have two subspaces of R^2: U = {(x,y) in R^2 : y + 2x = 0} W = {(x,y) in R^2 : y - 3x = 0} and I wanted to geometrically (and algebraically) represent their...
  8. A

    Subspaces and eigenvalues

    Given a square matrix, if an eigenvalue is zero, is the matrix invertible? I am inclined to say it will not be invertible, since if one were to do singular value decomposition of a matrix, we would have a diagonal matrix as part of the decomposition, and this diagonal matrix would have 0 as an...
  9. G

    Splitting Polynomials into Even and Odd Parts: A Unique Direct Sum Decomposition

    1. Let \mathbb{R}[x]_n^+ and } \mathbb{R}[x]_n^- denote the vector subspaces of even and odd polynomials in \mathbb{R}[x]_n Show \mathbb{R}[x]_n=\mathbb{R}[x]_n^+ \oplus\mathbb{R}[x]_n^- 3. For every p^+(x) \in \mathbb{R}[x]_n^+ \displaystyle p^+(x)=\sum_{m=0}^n a_m x^m=p^+(-x)...
  10. J

    Find a basis for the following subspaces

    Homework Statement Homework Equations In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system".[1] In more general...
  11. N

    Is U + U' a subspace if U and U' are contained in W?

    If U, U ′ are subspaces of V , then the union U ∪ U ′ is almost never a subspace (unless one happens to be contained in the other). Prove that, if W is a subspace, and U ∪ U ′ ⊂ W , then U + U ′ ⊂ W . This seems fairly simple, but I am stuck on how to go about proving it.
  12. F

    What is the dimension of the subspace formed by two vectors in R3?

    Homework Statement http://img6.imageshack.us/img6/5017/69430037.th.png Uploaded with ImageShack.us The Attempt at a Solution a) http://www4d.wolframalpha.com/Calculate/MSP/MSP10519f5ffahf530d2ei00005867aii1c038ibgc?MSPStoreType=image/gif&s=29&w=178&h=56...
  13. A

    Matrix Subspaces: Does Set W = {X: AX=2X} Form a Subspace of M(2,1)?

    Homework Statement Let A be a fixed 2x2 matrix. Assuming that the set: W={X:AX=2X} has infinitly many solutions, determine whether it is a subspace of M(2,1) Homework Equations To determine whether a set is a subspace i need to prove that there is a zero vector, that it is closed under...
  14. H

    Proving Subspaces of Finite-Dimensional Vector Spaces

    1) How to show that if W is a subspace of a finite-dimensional vector space V, then W is finite-dimensional and dim W<= dimV. 2) How to show that if a subspace of a finite-dimensional vector space V and dim W = dimV, then W = V. 3) How to prove that the subspace of R^3 are{0}, R^3 itself...
  15. J

    which of the following are linear subspaces ?

    Homework Statement For each set below, which of the following sets are real linear subspaces (where addition and scalar multiplication are defined in the usual way for these sort of objects)? Justify your answers with an argument in which real linear space it is included, why it is closed...
  16. mccoy1

    Vector Spaces: Determining Subspaces in R^3 and R^2

    Homework Statement Which of the following are a vector spaces? (a)R = {(a,b,c) 〖 ∈R〗^3 │ a+b=0 and 2a-b-c=0} (b)S = {(a,b)∈ R^2│ a^2=b^2 }..., can I say a =b which will make things simpler? (c)T = {(f∈P_2 (R)│ f(1)=f(0)+1} Homework Equations The Attempt at a Solution...
  17. mccoy1

    Finding a Non-Trivial Quadratic in the Intersection of Two Subspaces

    Homework Statement I'm given two subspaces L and K of P2 (R) are given by L = { f(x) : 19f(0)+f ' (0) = 0 } K = { f(x) : f(1) = 0 }. Obtain a non-trivial quadratic n = ax2 + b x +c such that n is element of the intersetion of L and K. Homework Equations The...
  18. D

    Dimension of The Intersection of Subspaces

    Homework Statement If V and W are 2-dimensional subspaces of \mathbb{R}^{4}, what are the possible dimensions of the subspace V \cap W? (A). 1 only (B) 2 only (C) 0 and 1 only (D) 0, 1, 2 only (E) 0,1,2,3, and 4 Homework Equations dim(V + W) = dim V + dim W - dim(V \cap W) dim (V + W) \leq...
  19. F

    Linear Algebra, subspaces and row reducing

    Homework Statement This is just a conceptual question Whenever you are asked for a basis for the subspace spanner by some set of vectors, is that the same as asking the basis that forms the column space of that matrix? Are the dimension for that subspace the same as the column space...
  20. W

    Help with Understanding Locally Compact Spaces & Subspaces

    hi.. how can we say a compact space automatically a locally compact? how subspace Q of rational numbers is not locally compact? am not able to understand these.. can anyone help me?
  21. T

    What is an example of a separable Hausdorff space with a non-separable subspace?

    Homework Statement Give an example of a separable Hausdorff space (X,T) with a subspace (A,T_A) that is not separable. The Attempt at a Solution well since a separable space is one that is either finite or has a one-to-one correspondence with the natural numbers, the separable...
  22. M

    Linear algebra: subspaces, linear independence, dimension

    Homework Statement 1. Consider three linearly independent vectors v1, v2, v3 in Rn. Are the vectors v1, v1+v2, v1+v2+v3 linearly independent as well? 2. Consider a subspace V of Rn. Is the orthogonal complement of V a subspace of Rn as well? 3. Consider the line L spanned by [1 2...
  23. M

    Unique Subspaces for Vector Space V in R3

    Homework Statement If R3 is a vector space and V = (x,x,0) is a subspace, find unique subspaces W1 and W2 such that R3 = V ⨁ W1 = V ⨁ W2 Homework Equations The Attempt at a Solution Assuming R3 = (x,y,z) - please correct me if I'm wrong somehow - then I could pick a W1 like...
  24. T

    Linear Algebra: Subspaces

    Homework Statement Determine in each case below whether U is a subspace V. If it is, verify all conditions for U to be a subspace, and if not, state a condition that fails and give a counter-example showing that the condition fails. There are parts (a) through (f) to this question. I am...
  25. B

    Simple Question on Homotopy Equiv. and Contractible Subspaces

    Hi, everyone: I have not found a clear explanation for this: Are any two contractible subspaces A,B<X homotopy-equivalent to each other? Clearly, homotopy equivalence is an equivalence relation. BUT: what if A<X is H-E (Homotopy-Equiv.) to a point p in X, while...
  26. F

    Distance between Compact Subspaces: Proving Existence of Minimum Distance

    Homework Statement For a metric space (M,d) and two compact subspaces A and B define the distance d(A,B) between these sets as inf{d(x,y): x in A and y in B}. Prove that there exists an x in A and a y in B such that d(x,y)=d(A,B). Homework Equations The Attempt at a Solution I...
  27. I

    How do I find the orthonormal basis for the intersection of subspaces U and V?

    Homework Statement Hi, i am trying to do the question on the image, Can some one help me out with the steps. [PLAIN]http://img121.imageshack.us/img121/6818/algebra0.jpg Solution in the image is right but my answer is so off from the current one. Homework Equations The...
  28. M

    Finding the intersection of subspaces, and addition of subspaces

    Heres the question: Let {u,v,w} be a linearly independent set of vectors of R^4. Let E = span{u,2v} and F=span{w,v}. Find EnF and E + F. i really have no idea other than i guess if 1/2u=w and v=v, then the EnF can be defined by that, but I'm not sure if that is right! :(
  29. I

    Subspaces of polynomials with degree <= 2

    Homework Statement Which of these subsets of P2 are subspaces of P2? Find a basis for those that are subspaces. (Only one part) {p(t): p(0) = 2} Homework Equations The Attempt at a Solution So, I know the answer is that it's not a subspace via back of the book, but I don't...
  30. T

    Direct Sum of Vector Subspaces: Exploring the Relationship between U, W, and V

    Homework Statement [PLAIN]http://img571.imageshack.us/img571/1821/subspaces.png Homework Equations The Attempt at a Solution Is my solution correct?: For a,b\in \mathbb{C} let A=\begin{bmatrix} a \\ a \\ 0 \end{bmatrix}\in U and B=\begin{bmatrix} 0 \\ b \\ b...
  31. L

    Determine wether or the following subsets are subspaces of F

    Homework Statement Let F be the vector space (over R) of all functions f : R−R. Determine whether or not the following subsets of F are subspaces of F: Homework Equations 1. S1 = {f e F|f(−3) = 0 and f(10) = 0}; 2. S2 = {f e F|f(−3) = 0 or f(10) = 0}. The Attempt at a Solution I...
  32. B

    Vector space and number of subspaces

    Homework Statement How many two dimensional subspaces does (F_3)^4 have? The attempt at a solution I chose an arbitrary basis so B = (v1,v2,v3,v4) for (F_3)^4 and then basically did 4C2 = 6 so it has 6 subspaces with dimension 2. However, thinking over this problem I've realized that I'm...
  33. W

    Distance between two linear subspaces

    suppose v1 and v2 are two linear subspaces of a linear subspace v is there any measure of the distance between the two subspaces? in two dimensional complex space, i think the distance between x and y axes is the maximum possible value. Intuitively, if two subspaces are orthogonal to each...
  34. N

    (algebra) Proving subspaces- functions

    Homework Statement Is U={f E F(\left|a,b\right|) f(a)=f(b)} a subspace of F(\left| a,b \right|) where F(\left| a,b \right|) is the vector space of real valued functions defined on the interval [a,b]?Homework Equations I know in order for something to be a subspace there are three conditions...
  35. O

    Proving Inclusion of Vector Subspaces in W: A Scientific Approach

    If U and V are vector subspaces of W and if U union V is also a subspace of W, how would i show that either U or V is contained in the other?
  36. N

    Subspace & Span in Rn: Definition & Examples

    Hello! Just a quick question. Is the following okay?: The span of a set of vectors corresponds to a subspace in Rn. But the span of a set of vectors can also be ALL of Rn, does that mean all of Rn can be considered a subspace? Or does it mean the first definiton is not entirely correct, and...
  37. estro

    Linear algebra - subspaces

    Suppose V is a vector space over R and U_1, U_2, W are subspaces of V. Prove or contradict: 1. W \cap (U_1+U_2) = (W \cap U_1)+(W \cap U_2) 2. If\ U_1 \oplus W = U_2 \oplus W\ then\ U_1=U_2 I'm not sure how to approach this problem, and will appreciate any guidance. Thanks! (For the...
  38. A

    Finding Orthonormal Basis of Hilbert Space wrt Lattice of Subspaces

    I have a Hilbert space H; given a closed subspace U of H let PU denote the orthogonal projection onto U. I also have a lattice L of closed subspaces of H, such that for all U and U' in L, PU and PU' commute. The problem is to find an orthonormal basis B of H, such that for every element b of B...
  39. E

    Proof involving subspaces of finite-dimensional vector spaces

    This is an exercise in a linear algebra textbook that I initially thought was going to be easy, but it took me a while to make the proof convincing. Prove: Any subspace of a finite-dimensional vector space is finite-dimensional. Here's my attempt. I am not sure about some details and I'm...
  40. jinksys

    Lin Algebra - Find a basis for the given subspaces

    Find a basis for the given subspaces of R3 and R4. a) All vectors of the form (a, b, c) where a =0. My attempt: I know that I need to find vectors that are linearly independent and satisfy the given restrictions, so... (0, 1, 1) and (0, 0, 1) The vectors aren't scalar multiples of...
  41. Telemachus

    What is the solution to finding spanned subspaces?

    Hi, I'm trying to solve this exercise. Homework Statement Given the following subsets of \mathbb(R)^3 find the subspaces generated by them: \{(1,0,-2),(-1,0,2),(3,0,-1),(-1,0,3)\} The Attempt at a Solution I've tried to solve the linear dependence, so I've made the system: \begin{Bmatrix}...
  42. H

    T-Cyclic Subspace Generated by Z Using T(f) = f' + 2f in P1(\Re)

    Homework Statement Find the T-Cyclic subspace generated by Z. V = P1(\Re) T(f) = f' +2f and Z = 2xHomework Equations The Attempt at a Solution so T(1,0) = 2 and T(0,1) = 1 + 2x so [T]_{}\beta = ( 2 1 0 2 ) So T-cyclic subspace generated by 2x = { 2x, 2 + 4x } ?
  43. H

    T-Invariant Subspaces: Proving W is T-Invariant for E_{\lambda}

    Homework Statement Show that W is a T-invariant subspace of T for: W = E_{\lambda}Homework EquationsThe Attempt at a Solution Ok, so I know that I need to show that T maps every element in E_{\lambda} to . E_{\lambda} = N(T-\lambdaI) so T must map every eigenvector related to \lambda to...
  44. J

    Restriction of SO(N) to 2 dim subspaces

    If N>2 and A\in\textrm{SO}(N) are arbitrary, does there exist subspaces V_1,V_2\subset\mathbb{R}^N such that V_1+ V_2 = \mathbb{R}^N,\quad\quad \textrm{dim}(V_1)=2,\quad \textrm{dim}(V_2)=N-2 and such that the restriction of A to V_1 belongs to \textrm{SO}(2), and that the restriction of A...
  45. N

    Proof that W Contains Span of Set S in Vector Space V

    Let S= { v1, v2, v3...vn} be a set of vectors in a vector space V and let W be a subspace of V containing S show W contains span S. Span is the smallest subspace (w) of vector space V that contains vectors in S if a and b are two vectors in subspace C, then they are linear combinations...
  46. C

    Proof about bases for subspaces

    Homework Statement Prove that all bases for subspace V of R\hat{}N contain the same number of elements. Homework Equations The Attempt at a Solution I have absolutely no idea where to start this proof. Do I need to do something with finding an equation of the subspace, or not?
  47. H

    Find Basis for R4 T-Cyclic Subspace Generated by e1

    Homework Statement For each linear operator T on the vector space V, find an ordered basis for the T-Cyclic subspace generated by the vector z. a) V = R4, T(a+b,b-c,a+c,a+d) and z= e1 Homework Equations Theorem: Let T be a linear operator on a finite dimensional vector space V, and let...
  48. H

    Prove: Invariant Subspaces are g(T)-Invariant

    Homework Statement Let T be a linear operator on a vector space V and let W be a T-Invariant subspace of V. Prove that W is g(T)-invariant for any polynomial g(t). Homework Equations Cayley-Hamilton Theorem? The Attempt at a Solution Im not sure how to begin. Ok so g(t) is the...
  49. K

    Subspaces and Orthogonality

    Ok so I've been working on this problem and I'm really having some struggles grasping it. Here it is: Let W be some subspace of Rn, let WW consist of those vectors in Rn that are orthognoal to all vectors in W. 1) Show that WW is a subspace of Rn? So for this part I'm thinking that...
  50. mnb96

    Distance between subspaces

    Hello. Let´s suppose we are given two subspaces of \mathbb{R}^n that have dimension k, where 1\leq k<n. I think they are called grassmanians. How can I compute a "distance" between two different k-subspaces? my attempt to a solution: As a toy example, for n=2 and k=1 we can use the...
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