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. kostoglotov

    Simple fundamental subspaces problem stumping me

    So the matrix is just a row vector \begin{bmatrix}3 & 4 & 0\end{bmatrix} My problem, is that I get the nullspace as having to 2 dimensions, and the row space as having 2 dimenions, but that adds up to 4 dimensions, when it should add up to three. What simple thing am I missing? Null space...
  2. kostoglotov

    Row and null complements of x; need clarity....

    I've managed to distill the rambling into just this question, posted here and at the end of my digressive thoughts as well: "Will we always be able to split x up in such a way that we have a nullspace component and a non-row space component?" Take a matrix A = \begin{bmatrix}1 & 2\\ 3 &...
  3. Andrew Pierce

    Determining subspaces for all functions in a Vector space

    Homework Statement First, I'd like to say that this question is from an Introductory Linear Algebra course so my knowledge of vector space and subspace is limited. Now onto the question. Q: Which of the following are subspaces of F(-∞,∞)? (a) All functions f in F(-∞,∞) for which f(0) = 0...
  4. RJLiberator

    Check my proof for quality (Inner Product Space / Subspaces)

    Homework Statement S = a non-empty set of vecotrs in V S' = set of all vectors in V which are orthogonal to every vector in S Show S' = subspace of V Homework Equations Subspace requirements. 1. 0 vector is there 2. Closure under addition 3. Closure under scalar multiplication The Attempt at...
  5. yango_17

    Checking if sets are subspaces of ##\mathbb{R}^{3}##

    Homework Statement Is the set ##W## a subspace of ##\mathbb{R}^{3}##? ##W=\left \{ \begin{bmatrix} x\\ y\\ z \end{bmatrix}:x\leq y\leq z \right \}## Homework EquationsThe Attempt at a Solution I believe the set is indeed a subspace of ##\mathbb{R}^{3}##, since it looks like it will satisfy...
  6. kostoglotov

    Help: All subspaces of 2x2 diagonal matrices

    The exercise is: (b) describe all the subspaces of D, the space of all 2x2 diagonal matrices. I just would have said I and Z initially, since you can't do much more to simplify a diagonal matrix. The answer given is here, relevant answer is (b): Imgur link: http://i.imgur.com/DKwt8cN.png...
  7. B

    MHB Interpreting U={A|A^2=A, A is in M22}: Not a Subspace of M22

    How do I interpret the following: U={A|A^2=A, A is an element of M22} is not a subspace of M22. I don't quite understand what it's asking in terms of A^2=A. Thanks.
  8. B

    MHB Proving Vector Subspace $U \notin R^3$

    Hi All, How do I prove that $U=\{(x,y,z)|x \mbox{ is an integer}\}$ is not a subspace of $R^3$? I understand that I have to show $U$ is closed or not closed under vector addition and scalar multiplication but I'm unsure how I represent $x$ as an integer. I would say: let $V=\{v=(x,y,z) \in...
  9. G

    Common supplementary subspaces

    Homework Statement Let ##E## be a finite dimensional vector space, ##A## and ##B## two subspaces with the same dimension. Show there is a subspace ##S## of ##E## such that ##E = A \bigoplus S = B \bigoplus S ## Homework Equations [/B] ##\text{dim}(E) = n## ##\text{dim}(A) = \text{dim}(B) = m...
  10. S

    Proving Subspaces of Vector Spaces: Evaluating A Vector x

    Homework Statement How would one determine if a vector space is a subspace of another one? I think that the basis vectors of the subspace should be able to be formed from a linear combination of the basis vectors of the vector space. However, that doesn't seem to be true for this question: Let...
  11. I

    Vector space, linear transformations & subspaces

    Homework Statement Let V be a vector space over a field F and let L and M be two linear transformations from V to V. Show that the subset W := {x in V : L(x) = M(x)} is a subspace of V .The Attempt at a Solution I presume it's a simple question, but it's one of those where you just don't...
  12. I

    Field axioms, subspaces

    Homework Statement Let F_{2} = {0, 1} denote a field with 2 elements. Let V be a vector space over F_{2}. Show that every non-empty set W of V which is closed under addition is a subspace of V. The Attempt at a Solution subspace axioms: 0 elements, closed under scalar multiplication, closed...
  13. ElijahRockers

    Inner Product Space - Pythagorean?

    Homework Statement Let ##V## be an inner product space and let ##V_0## be a finite dimensional subspace of ##V##. Show that if ##v ∈ V## has ##v_0 = proj_{V_0}(v)##: ||v - vo||^2 = ||v||^2 - ||vo||^2 Homework Equations General inner product space properties, I believe. The Attempt at a...
  14. K

    MHB Show non-degenerate form of subspaces sum

    Let (E, d) be nonzero bilinear space over K and place conditions: d(x,y) = d(y,x) \\ d(x,y) = - d(y,x) for every x,y \in E. Show that: if E_1 and E_2 are singular (degenerate?) bilinear subspaces relative with d ( (E_1,d|(E_1 \times E_1) and (E_2,d|(E_2 \times E_2) are singular (degenerate?)...
  15. C

    Linear subspaces and dimensions Proof

    Homework Statement Let Pn denote the linear space of all real polynomials of degree </= n, where n is fixed. Let S denote the set of all polynomials f in Pn satisfying the condition given. Determine whether or not S is a subspace of Pn. If S is a subspace, compute dim S. The given condition if...
  16. M

    Maximal Ideals and Maximal Subspaces in normed algebras

    Homework Statement Let A be a unital commutative Banach Algebra and I a maximal ideal of A. Prove that I is a maximal subspace of A. Is this result still valid if A is not Banach or commutative or unital?Homework Equations The first part is pretty easy: Maximal ideals are of the form kerτ for...
  17. BiGyElLoWhAt

    Another question about transforms and subspaces

    Homework Statement A: Let ##T## be the linear function ##T####:\mathbb{R}^3→\mathbb{R}^1## defined as ##T####(x,y,z) = x-3y+z##. The nullspace of T is a 2 dimensional subspace of ##\mathbb{R}^3## (a plane through the origin). Give an example of the basis of this subspace ##\{...
  18. T

    Union of subspaces: proving a biconditional statement

    Homework Statement Let ##W_1## and ##W_2## be subspaces of a vector space ##V##. Prove that ##W_1 \cup W_2## is a subspace of ##V## if and only if ##W_1 \subseteq W_2## or ##W_2 \subseteq W_1##. Homework Equations A subset ##W## of a vector space ##V## is a subspace of ##V## provided...
  19. PsychonautQQ

    Polynomials in n variables subspaces and subrepresentations

    Homework Statement Trying to make sense of my notes... "A polynomial in n variables on an n-dimensional F-vector space V is a formal sum of the form: p(x)= ∑(C_i)x^β" so basically can somebody help me understand how polynomials represent vector spaces? Whatever degree the polynomial is...
  20. F

    Understanding Subspaces in Rn: Exploring the Role of Orthonormal Bases

    Hello everyone, I have a theoretical question on subspaces. Consider the space Rn. The zero vector is indeed a subspace of Rn. However, if I am not mistaken, the zero vector has no orthonormal basis, even though it is a subspace. I thought all subspaces have an orthonormal basis (or is it...
  21. Seydlitz

    Showing that V is a direct sum of two subspaces

    Hi guys, I have this general question. If we are asked to show that the direct sum of ##U+W=V##where ##U## and ##W## are subspaces of ##V=\mathbb{R}^{n}##, would it be possible for us to do so by showing that the generators of the ##U## and ##W## span ##V##? Afterwards we show that their...
  22. N

    Interesting Subspaces of ##L^p## Spaces

    ##C_0=\{f\in L^p: f(x)\rightarrow 0 ## as ## x\rightarrow infinity\}## This is an interesting subspace because it is the subspace of ##L^p## in which the momentum operator from physics is self adjoint. It seems that there should be more to be said about the importance of ##C_0## though...
  23. C

    MHB Invariant subspaces of representations

    Let $\varrho :\mathbb{Z}\rightarrow GL_3(\mathbb{R})$ be the representation given by $\varrho (n)=A^n$ where A=$\begin{pmatrix} 2 & 5 & -1 \\ 2 & \frac{5}{2} & \frac{11}{2} \\ 6 & \frac{-2}{2} & \frac{3}{2} \\ \end{pmatrix}$ Does ρ have any 1-dimensional invariant subspaces? Do I have to...
  24. P

    Vector Subspaces Homework: Is (x,y,z) a Subspace of R^3?

    Homework Statement (x,y,z) where 2x + 2y + z = 1 Is this set a subspace of R^3? The Attempt at a Solution I am thinking it is not since it does not contain the origin since 2(0)+2(0) + 0 = 1 0 != 1 (!= means not equal) Am I right? I am kind of having trouble with this part of...
  25. C

    MHB Show all invariant subspaces are of the form

    [solved] show all invariant subspaces are of the form i don't even know how to begin (Angry) C_x is a subspace spanned by x that belongs to V C_x = {x, L(x), L^2(x),...} edit: SOLVED
  26. Sudharaka

    MHB Collection of Subspaces of a Vector Space

    Hi everyone, :) Here's a question I am struggling with recently. Hope you can give me some hints or ideas on how to solve this. Question: If the collection of subspaces of the \(K\)-vector space \(V\) satisfies either distributive law \(A+(B\cap C)=(A+B)\cap (A+C)\) or \(A\cap (B+C)=(A\cap...
  27. J

    Calculating the Intersection of Subspaces in Vector Spaces

    Given two subspaces of the vector space of all polynomials of at most degree 3 what is the general method to calculate the intersection of the two subspaces?
  28. NATURE.M

    Proving the subspaces are equal

    If I want to show two orthogonal subsets S_{1} and S_{2} of ℝ^{n} both span the same subspace W of ℝ^{n} does it suffice to show that S_{1}\subsetS_{2} and that S_{2}\subsetS_{1}, thus showing S_{1} = S_{2} \Rightarrow they span the same space. If there's a better method, I'd like to know...
  29. T

    MHB Find T cyclic operator that has exactly N distinct T-invariant subspaces

    Let T be a cyclic operator on $R^3$, and let N be the number of distinct T-invariant subspaces. Prove that either N = 4 or N = 6 or N = 8. For each possible value of N, give (with proof) an example of a cyclic operator T which has exactly N distinct T-invariant subspaces. Am I supposed to...
  30. M

    Sum of two closed subspaces in a Banach space

    Homework Statement . Let ##E## be a Banach space and let ##S,T \subset E## two closed subspaces. Prove that if dim## T< \infty##, then ##S+T## is also closed. The attempt at a solution. To prove that ##S+T## is closed I have to show that if ##x## is a limit point of ##S+T##, then ##x \in...
  31. C

    Linear algebra question about subspaces

    Homework Statement This is probably a very dumb question, but I just can't wrap my head around what I'm supposed to be doing. The question is: "Determine whether the set is a subspace of R3: All vectors of the form (a,b,c) where a = 2b + 3c" Homework Equations u + v is an...
  32. 1

    How do I prove the subspace property for M and N in Linear Algebra?

    Homework Statement Let V = V1 + V2, where V1 and V2 are vector spaces. Define M ={(x1, 0vector2): x1 in V1} and N = {(0vector1, x2) : x2 in V2 0vector 1 is the 0v of V1 and 0vector is the 0v of V2 and 0v is 0 vector of V a) prove hat both M and N are subspace of V b) show that M n N...
  33. W

    How to sample subspaces uniformly

    i need to sample the N-dimensional subspaces of a M-dimensional linear space over C uniformly. That is, all subspaces are sampled with equal probability how should i do it? would this work? First generate a M*N matrix, the real and imaginary parts of each element is sampled from the...
  34. Fernando Revilla

    MHB Jae 's question at Yahoo Answers (Intersection of subspaces)

    Here is the question: Here is a link to the question: Intersection of subspaces P1 and P2? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  35. S

    Write down orthonormal bases for the four fundamental subspaces [ ]

    Write down orthonormal bases for the four fundamental subspaces [...]" Homework Statement Problem: Write down orthonormal bases for the four fundamental subspaces of A = matrix([1,2],[3,6]]). (1 and 2 are on the first row whereas 3 and 6 are on the second row.) Solution: A =...
  36. W

    Is W a Subspace of All Ordered Pairs?

    Homework Statement Let W be the set of all ordereed pairs of real numbers, and consider the following addition and scalar multiplication operations on U=(u1,u2) and V=(v1,v2) U+V is standard addition but kU=(0, ku2) Homework Equations Is W closed under scalar multiplication? The Attempt at a...
  37. S

    Subspaces of ℝ^3 and their bases

    Homework Statement Determine whether each of the following is a subspace of ℝ3, and if so find a basis for it: a) The set of all vectors (x, y) b) The set of all vectors of the form (sin2t, sintcost, 3sin2t) Can someone please explain to me how you determine whether these are...
  38. M

    What are the subspaces of ℝ, ℝ^2, and ℝ^3?

    Homework Statement Find all subspaces of the vector spaces: (ℝ+,.) , (ℝ^2 +,.) , (ℝ^3 +,.) The Attempt at a Solution For ℝ the only subspace i can think of is {0} For ℝ^2 if found {0} R^2 itself and any set of the form L=cu for u≠0. For ℝ^3 if found those of R^2 plus R^3 . Are...
  39. M

    Linear algebra question: Orthogonal subspaces

    Homework Statement For each of the following matrices, determine a basis for each of the subspaces N(A) A=[3 4] [ 6 8]Homework Equations The Attempt at a SolutionSo reducing it I got [1 4/3] [0 0] I know x2 is a free variable I set x2 = to β and found...
  40. M

    Prove that ℝ has no subspaces except ℝ and {0}.

    Prove that ℝ has no subspaces except ℝ and {0}.
  41. G

    Subspaces and interiors of metric spaces problem.

    Homework Statement If S is a subspace of the metric space X prove (intxA)\capS\subsetints(A\capS) where A is an element of ΩX(Open subsets of X) The Attempt at a Solution So intxA=\bigcupBd(a,r) where d is the metric on X and the a's are elements of A and I think...
  42. G

    Discrete topology and discrete subspaces

    Homework Statement If A is a subspace of X and A has discrete topology does X have discrete topolgy?Also if X has discrete topology then does it imply that A must have discrete topology? The Attempt at a Solution My understanding of discrete topology suggests to me that if A is discrete it...
  43. M

    Are These Subsets of R3 Subspaces?

    Homework Statement Which of the following subsets of R3 are subspaces? The set of all vectors of the form (a,b,c) where a, b, and c are... Homework Equations 1. integers 2. rational numbers The Attempt at a Solution I think neither are subspaces. IIRC, the scalar just needs to be...
  44. M

    Determining if sets are subspaces of vector spaces

    Homework Statement Are the following sets subspaces of R3? The set of all vectors of the form (a,b,c), where 1. a + b + c = 0 2. ab = 0 3. ab = ac Homework Equations Each is its own condition. 1, 2 and 3 do not all apply simultaneously - they're each a separate question. The...
  45. M

    Scalars and determining subspaces

    To determine if a subset of a vector space is a subspace, it must be closed under addition and scalar multiplication. As far as I can tell, this means adding two arbitrary vectors in the subset and having the sum be within the subset. But...can the scalar be any number? Is there any limitation?
  46. S

    T-invariant subspaces

    Homework Statement Prove that the intersection of any collection of T-invariant subspaces of V is a T-invariant subspace of V.Homework Equations The Attempt at a Solution Let W1 and W2 be T-invariant subspaces of V. Let W be their intersection. If v\inW, then v\inW1 and v\inW2. Since v\inW1...
  47. S

    Prove the intersection of two orthogonal subspaces is {0}

    Homework Statement Let A and B be two orthogonal subspaces of an inner product space V. Prove that A\cap B= \{ 0\}. Homework Equations The Attempt at a Solution I broke down my proof into two cases: Let a\in A, b\in B. Case 1: Suppose a=b. Then \left\langle a,b \right\rangle =...
  48. H

    Proofs of dimensions and subspaces check

    Hi, I'd be grateful if someone could tell me whether these proofs I've done are correct or not. Thanks in advanced. Let V be an n-dimensional vector space over \mathbb{R} Prove that V contains a subspace of dimension r for each r such that 0 \leq r \leq n Since V is n-dimensional...
  49. H

    Dimension proof of the intersection of 3 subspaces

    Homework Statement Assume V = \mathbb{R}^n where n \geq 3. Suppose that U,W,X are three distinct subspaces of dimension n-1; is it true then that dim(U \cap W \cap X) = n-3? Either give a proof, or find a counterexample.The Attempt at a Solution The question previous to this was showing that...
  50. P

    Condition for equality between subspaces.

    Hi, Homework Statement What would be the/a condition on vectors in K so that V=W, where V is a vector space which K={v1,v2,v3,v4} spans, and W is a subspace of V defined thus: W=Sp{v1+v2,v2+v3,v3+v4,v4+v1} Homework Equations The Attempt at a Solution I believe V would be equal...
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