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. K

    What is the Difference Between a Lie Subalgebra and a Subspace?

    I have a question about Lie subalgebra. They say "a Lie subalgebra is a much more CONSTRAINED structure than a subspace". Well, it seems subtle, and I find this very tricky to follow. Can anyone explain this with concrete examples? If my question is not clear, please tell me so, I will...
  2. K

    Subspace Span Determination for Vector y in R^4 using Augmented Matrix

    Homework Statement For each s \inR determine whether the vector y is in the subspace of R^4 spanned by the columns of A where y= 6 7 1 s and A = 1 3 2 -1 -1 1 3 8 1 4 9 3 Homework Equations The Attempt at a Solution Can i do this by making an...
  3. S

    Finding basis for subspace

    use rowspace/colspace to determine a basis for the subspace of R^n spanned by the given set of vectors: {(1,-1,2),(5,-4,1),(7,-5,-4)} *note: the actual instructions are to use the ideas in the section to determine the basis, but the only two things learned in the section are rowspace and...
  4. F

    Proving Subspace Membership in R^2 using Vector Dot Product

    The question I am looking at asks, Let a be the vector (-1,2) in R^2. Is the set S = { x is in R^2 | x dot a = 0} a subspace? --> x and a are vectors... Can anyone explain how to show this? I was thinking that since the zero vector is in R^2, this must also be a subspace...
  5. K

    Subspace criteria applied to square matrices, proof help

    Homework Statement Let n \geq2. Which of the conditions defining a subspace are satisfied for the following subsets of the vector space Mnxn(R) of real (nxn)-matrices? (Proofs or counterexamples required. There are three subsets, i will start with the one where The subset V is that of...
  6. K

    Subspaces in Vector Spaces over F2

    Homework Statement Let (F2) ={0,1} denote the field with 2 elements. i) Let V be a vector space over (F2) . Show that every non empty subset W of V which is closed under addition is a subspace of V. ii) Write down all subsets of the vector space (F2)^2 over (F2) and underline those...
  7. A

    Subspace of P4: Polynomials of Even Degree

    Homework Statement Determine whether the following is a subspace of P_{4}_ (a) The set of polynomials in P_{4} of even degree. Homework Equations P_{4} = ax^{3}+bx^{2}+cx+d The Attempt at a Solution (p+q(x)) = p(x) + q(x) (\alpha p)(x) = p(\alpha x) If p and q are both of...
  8. U

    Reaching all of R^N by rotations from a linear subspace

    Hi all, I have a problem related to quantum mechanical description of vibrational motion in molecules. I would like, for efficiency, to integrate over symmetries (global rotations) of the molecule. I would like to prove (or disprove) that all points in R^N can be reached by rotations from a...
  9. P

    Vector Subspace U+W = R^4: Solving with Homework Equations

    Homework Statement Are the vector subspaces U={(x,y,0,0) | x+2y=0} and W= {(0,0,z,t | z+t=0} from R^4 stand for U+W = R^4Homework Equations The Attempt at a Solution Can somebody explain, how will solve this task. I have no idea, how they do in my book. Thanks.
  10. P

    How do you determine the intersection of two vector subspaces?

    Hi! I just want to ask you, what is the principle of finding section between two vector subspaces. Let's say: U={(a,b,0) | a, b \in R} and W={(a,b,c) | a+b+c=0, a,b,c \in R} U, W are vector subspaces from R\stackrel{3}{}. P.S this is not homework question, just example, for my better...
  11. quasar987

    Two closed subspace whose sum is not closed?

    What would be an example of two closed subspaces of a normed (or Banach) space whose sum A+B = {a+b: a in A, b in B} is not closed? I suppose we would have to look in infinite dimensional space to find our example, because this is hard to imagine in R^n!
  12. T

    Why Does a Subset of a Vector Space Need the Zero Vector to Be a Subspace?

    I am curious as to why a subset of a vector space V must have the vector space V's zero vector be the subsets' zero vector in order to be a subspace. Its just not intuitive.
  13. B

    Linear Algebra Subspace Basis Problem

    1. The set of all traceless (nxn)-matrices is a subspace sl(n) of (bold)K^(nxn). Find a basis for sl(n). What is the dimension of sl(n)? Not sure how to go about finding the basis. I know a basis is a list of vectors that is linearly independent and spans. and for the dimension of sl(n), is...
  14. N

    Finding a basis for a subspace

    [SOLVED] Finding a basis for a subspace Homework Statement We have a subspace U in R^3 defined by: U = {(x_1 , x_2 ; x_3) | x_1 + 2*x_2 + x_3 = 0 }. Find a basis for U. The Attempt at a Solution We have the following homogeneous system: (1 2 1 | 0). From this I find the...
  15. quasar987

    What is an example of a non-closed subspace in a normed space?

    What would be an example of a not (topologically) closed subspace of a normed space?
  16. S

    Finding a Subspace W of R^4 for Direct Sum V(+)W

    V is a subspace of R^4 V={(x, -y, 2x+y, x-2y): x,y E R} 1) extend {(2,-1,5,0)} to a basis of V. 2) find subspace W of R^4 for which R^4= direct sum V(+)W. solution: 1)the dimension of V is 2.therefore i need to add one more vector to (2,-1,5,0). the 2nd vector is (1,0,2,1)...
  17. E

    Linear combination subspace help

    Homework Statement In a space V^{n} , prove that the set of all vectors \left\{|V^{1}_{\bot}> |V^{2}_{\bot}> |V^{3}_{\bot}> ... \right\} orthogonal to any |V> \neq 0 , form a subspace V^{n-1} Homework Equations The Attempt at a Solution I tried to make a linear combination from...
  18. T

    Quantum Foam: Could It Be Used for Subspace Communication?

    Could Quantum Foam be used as a type of subspace, like the kind used in Star Trek? Mainly for communication...
  19. E

    Proving W is a Subspace of V: Let u & v be Vectors in V

    Let u and v be (fixed) vectors in the vector space V. Show that the set W of all linear combinations au+bv of u and v is a subspace of V. I cannot prove the above proof properly. Can anyone help. -Thanks
  20. S

    Determine if H is a Subspace of Mnxn: Let A be an nxn Matrix

    Let A be an nxn matrix and let H= {B E Mnxn|AB=BA}. Determine if H is a subspace of Mnxn. This was a test question that I got incorrect. I didn't like the way my teacher proved this afterwards, they said it IS a subspace of Mnxn. Any help in explaining how it could be would be greatly...
  21. MathematicalPhysicist

    Proving Equality of Subspace Topologies: A Topological Lemma Approach

    Well it's not homewrok cause i don't need to hand this question in, this is why i decided to put it here. (that, and there isn't a topology forum per se, perhaps it's suited to point set topology so the set theory forum may suit it). Now to the question: Show that if Y is a subspace of X...
  22. E

    Is W a subspace of the vector space?

    W={(x1,x2,x3):x^{2}_{1}+x^{2}_{2}+x^{2}_{3}=0} , V=R^3 Is W a subspace of the vector space? from what i understand for subspace to be a subspace it has to have two conditions: 1.must be closed under addition 2.must be closed under multiplication so... I pick a vector s=(s1,s2,s3) and a second...
  23. E

    Proving Subspace in R^3 with Fixed Matrix A: W={x \in R^{3} :Ax=[^{1}_{2}]}

    Let A be a fixed 2x3 matrix. Prove that the set W={x \in R^{3} :Ax=[^{1}_{2}]} (2x1 matrix 1 on top 2 at the bottom) what does the information after the ":" mean? is it a condition? I don't understand this problem. Can anyone help me out?
  24. B

    Questions about the concept of subspace of linear transformation

    Hi all, I have some questions about the concept of subspace of linear transformation and its dimension, when I try to prove following problems: Prove T is a finite dimensional subspace of L(V) and U is a finite dimensional subspace of V, then T(U) = {F(u) | F is in T, u is in U} is a...
  25. C

    Vector subspace notation question

    when dealing with vector subspaces, an example of a question asks: if x=(a,a,b)T (T is really a superscript) is any vector in S, then... what does that T superscript notation mean? does that mean transpose? Thanks
  26. L

    Algebra- Vector ce and subspace

    Algebra- Vector space and subspace Homework Statement Here are some true or false statements given in my test. (a) R^2 is a subspace of R^3. (b) If {v1, v2, ..., vn} is a set of linearly dependent vectors, then it contains a zero vector. (c) If {v1, v2, ..., vn} is a spanning set, then...
  27. M

    Linear Algebra: Subspace Proof

    Prove that the intersection of any collection of subspaces of V is a subspace of V. Ok, I know I need to show that: 1. For all u and v in the intersection, it must imply that u+v is in the intersection, and 2. For all u in the intersection and c in some field, cu must be in the...
  28. D

    Subspace Problems: Which R^n*n Subsets are Subspaces?

    Homework Statement Which of the following subsets of R^n*n are in fact subspaces of R^n*n 1) The symmetric matrices 2) The diagonal matrices 3) The nonsingular matrices 4) The singular matrices 5) The triangular matrices 6) The upper triangular matrices 7) All matrices that commute...
  29. Q

    Is every subspace of a connected space connected?

    Is every subspace of a connected space connected?
  30. T

    Finding a Basis for the Subset of Polynomials Satisfying p(5)=0 | Exam Prep

    I am drawing some strange mental blank with one question in my final exam review. Homework Statement V is the set of all polynomials that are of the form p(x) = cx^2 + bx+a U is a subset of V where all members satisfy the equation p(5) =0 Find a basis for U. I am not sure why I...
  31. U

    Is the third condition necessary for a vector space to be considered a subspace?

    Anton, H. Elementary linear algebra (5e, page 156) says: If W is subset of a vector space V, then W is a subspace of V if and only if the following TWO conditions hold 1) If u and v are vectors in W, then u+v is in W 2) If k is any scalar and u is any vector in W, then ku is in W...
  32. M

    Vector Subspace or Linear Manifold.

    Is there any difference between a vector subspace and a linear manifold. Paul Halmos in Finite Dimensional Vector Spaces calls them the same thing. Hamburger and Grimshaw in Linear Trasforms in n Dimensional Vector Space does not use the word subspce at all. Planet Math says a Linear...
  33. 4

    Proving Theorem: Column Space of Matrix A is a Subspace of R^m

    How would I prove this theorem: "The column space of an m x n matrix A is a subspace of R^m" by using this definition: A subspace of a vector space V is a subset H of V that has three properties: a) the zero vector of V is in H. b) H is closed under vector addition. c) H is closed under...
  34. A

    What is the solution space and subspace of a given matrix?

    Homework Statement Let A = [2 -1 9] [1 0 5] [0 1 1] Find solution space W and prove that W is a subspace of R^3. Homework Equations The Attempt at a Solution rref= [1 0 5] [0 1 1] [0 0 0] So I know the row-echeleon form, which is what I suppose is the solution...
  35. A

    Vector Spaces, Dimension of Subspace

    Find the dimension of the subspace spanned by the vectors u, v, w in each of the following cases: i) u = (1,-1,2)^T v = (0,-1,1)^T w = (3,-2, 5)^T ii) u = (0,1,1)^T v = (1,0,1)^T w = (1,1,0)^T Right, how do I go about this, do I have to find the subspace first then do the dimension. Can...
  36. K

    Determinant, subspace, linear transformation

    I am having some trouble with the following linear algebra problems, can someone please help me? 1) Explain what can be said about det A (determinant of A) if: A^2 + I = 0, A is n x n My attempt: A^2 = -I (det A)^2 = (-1)^n If n is be even, then det A = 1 or -1 But what happens when n...
  37. R

    Is the Subset (b1, b2, b3) a Subspace of R3?

    Q: Is the subset a subspace of R3? If so, then prove it. If not, then give a reason why it is not. The vectors (b1, b2, b3) that satisfy b3- b2 + 3B1 = 0 ----------------------- My notation of a letter with a number to the right, (b1) represents b sub 1. Im having a problem on how...
  38. J

    Basis of a subspace of a vectorspace

    Is the basis for the subspace W of a vectorspace V spanned by the basis of the vectorspace V? If so how?
  39. E

    What are the possible values of dimS for a given vector x in R^4?

    Homework Statement choose x = (x1, x2, x3, x4) in R^4. It has 24 rearrangements like (x2, x1, x3, x4) and (x4, x3, x1, x2). Those 24 vectors including x itself span a subspace S. Find specific vectors x so that dimS is 0, 1, 3, 4 The Attempt at a Solution So, i thought of it this way: 24...
  40. E

    What is the smallest subspace of 3x3 matrices

    I want to confirm something: what is the smallest subspace of 3x3 matrices that contains all symmetric matrices and lower triangular matrices? - identity(*c)? because that is the only symmetric lower triangular i could think of... what is the largest subspce that is contained in both of...
  41. radou

    Show Existence of Basis for Vector Space V with No Elements from Subspace M

    Let V be a vector space over a field F, and M a subspace of V, where M is not {0}. I need to show there exists a basis for V such that none of its elements belong to M. Since M is a subspace of V, M must be a subset of V. If M = V, then there does not exist such a basis, so M must be a...
  42. JasonJo

    Finding a basis for a subspace

    Let U be a proper subspace of R^4 and let it be given by the equations: 1) x1+x2+x3+x4=0 2) x1-x2+2x3+x4=0 how do i find a basis for this subspace? I got that (0,1,2,0) is one of the basis vectors since x2=2x3, therefore whatever we pick for x2, x3 will be twice that value. i also...
  43. N

    What are the properties to prove a plane is a subspace of R^3?

    I have a question that states: Let a, b, and c be scalers such that abc is not equal to 0. Prove that the plane ax + by + cz = 0 is a subspace of R^3.
  44. E

    Subspace Questions: Checking 2 Sets in R^3

    I am not sure about these 2 whether they are subspaces or not (i do know how to check whether it is a subspace or not) subsets of R^3, subspace or not? 1.all combinations of (1,1,0) and (2,0,1) 2.plane of vectors (b1,b2,b3) that satisfy b3-b2+3b1 = 0 thanks for help.
  45. B

    Why R2 is not a subspace of R3?

    I think R2 is a subspace of R3 in the form(a,b,0)'.
  46. V

    Null subspace of a single vector

    Hi: was wondering if somebody can help me with this I came across in a paper. $e_k$ is a vector of $k$ $1's. M is a matrix of size n \times k. The author talks about projecting $M$ onto the null space of $e_k$. This is what confuses me. Which $x$ apart from the 0-vector solves e_kx=0...
  47. F

    Linear Algebra Test Prep: Determining Subspaces of Matnxn

    I'm preparing for an end of year linear algebra test which briefly covers things about subsets of Matnxn and their relations to subspaces of Matnxn; I found the following question: Which of the following subsets of Matn×n are subspaces of Matn×n? (i) Symmetric matrices (i.e. matrices A...
  48. S

    Calculating Subspace Dimension in R4: U = [a b c d] | a+b=c+d

    Find a basis and calculate the dimension of the following subspaces of R4 U = [a b c d] | a+b=c+d in R now a basis is supposed to be linear independant but that conditon on a,b,c,d makes this a bit strange can i just say d = a+b - c and then pick a basis as (1,0,0,1),(0,1,0,1),(0,0,1,-1)...
  49. S

    Can Spanning Subsets X and Y in Vector Space V Share No Common Vector?

    If X and Y are nonempty subsets of vector space V such taht span X = span Y = V , must there be a vector common to both X and Y? Justify. my intuition thinks there need not be one, but i can be wrong! to go about proving it is where the fun starts! a_{1} x_{1} + a_{2} x_{2} + ... + a_{n}...
  50. G

    Finding a basis for a subspace of P_2

    Let W=\lbrace p(x) \in P_{2} : p(2)=0\rbrace Find a basis for W. Since a basis must be elements of the set W we know that p(2)=0. So if p(x)=ax^2+bx+c, then p(x) = 4a+2b+c=0. Let c=t, b=s and s,t are real scalars. Then p(x) can be written as t(-\frac{1}{4} x^2+1)+s(-\frac{1}{2}x^2+x)...
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