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.

View More On Wikipedia.org
Recent contents Top users
  1. R

    Linear Algebra - subspaces of f in C[-1,1]

    Homework Statement Determine whether the following are subspaces of C[-1,1]: d) The set of functions f in C[-1,1] such that f(-1)=0 AND f(1)=0Homework Equations The Attempt at a Solution I did the question with 'OR', but I don't think I can find the functions. I am not sure I can use x2 here...
  2. G

    How to prove that the dim of two subspaces added together equals dim

    How to prove that the dim of two subspaces added together equals dim of their union plus 1 iff one space is a subest of the other In other words, subspaces: V, S of Vector space: W dim(V+S) = dim(V \cap S) +1 if V \subseteq S or S \subseteq V
  3. R

    Identifying Subspaces in R^2: A Linear Algebra Question

    Homework Statement Determine whether the following sets form subspaces of R^2 : a) {(x1,x2)T | x1*x2=0} b) {(x1,x2)T | x12=x22} c) {(x1,x2)T | |x1|=|x2| }Homework Equations The Attempt at a Solution My problem here is that I don't think I understand how the vectors look. for instance...
  4. J

    Subspaces and perpendiculuar subspaces

    Homework Statement How do you show that M double perp is a subset of M? Homework Equations The Attempt at a Solution My prof told me to try proving that M is a subset of M perp perp, then to use the facts that if M is a subspace of Rn then T(X) = projU(X) for all X in Rn. I'm not sure how to...
  5. J

    Linear Algebra: Polynomials subspaces

    U and W are subspaces of V = P3(R) Given the subspace U{a(t+1)^2 + b | a,b in R} and W={a+bt+(a+b)t^2+(a-b)t^3 |a,b in R} 1) show that V = U direct sum with W 2) Find a basis for U perp for some inner product Attempt at the solution: 1) For the direct sum I need to show that it...
  6. B

    Complete Metric Subspaces: Are These Metric Subspaces Complete?

    Homework Statement Determine whether the following metric subspaces are complete: a) the set E of sequences containing only entries 0 & 1 in (m,||\cdot||_{\infty}) b) the unit sphere in any Banach Space Homework Equations a) for x=\{\lambda_1,\lambda_2,\ldots,\lambda_n,\ldots \}...
  7. K

    Linear Algebra - Polynomial Subsets of Subspaces

    Homework Statement Which one of the following subsets of P_{2} (degree of 2 or below) are subspaces? a) a_{2}t^{2} + a_{1}t + a_{0}, where a_{1} = 0 and a_{0} = 0 b) a_{2}t^{2} + a_{1}t + a_{0}, where a_{1} = 2a_{0} c) a_{2}t^{2} + a_{1}t + a_{0}, where a_{2} + a_{1} + a_{0} = 2 Homework...
  8. S

    Really basic linear algebra: subspaces of F[a,b]

    Homework Statement Determine which of the following sets of functions are subsets of F[a,b] a) All functions f in F[a,b] for which f(a) = 0 b) All functions f in F[a,b] for which f(a) = 1 The Attempt at a Solution Ok so I am just learning about vector subspaces. After reading the...
  9. D

    (linear algebra) union of subspaces

    Eh, kind of stuck on this question. I need some suggestions on how to tackle the problem.. Homework Statement Let U and V be the subspaces of R_3 defined by: U = {x: aT * x = 0} and V = {x: bT * x = 0} (T means transpose) where a = [1; 1; 0] and b = [0; 1; -1] Demonstrate that...
  10. D

    Proof with intersection of subspaces

    Homework Statement Suppose L, M, and N are subspaces of a vector space. (a) Show that the equation L \cap (M+N) = (L \cap M)+(L \cap N) is not necessarily true. (b) Prove that L \cap (M+(L \cap N))=(L \cap M) + (L \cap N) Homework Equations N/A The Attempt at a Solution...
  11. T

    Prove sum of two subspaces is R^3

    How do you prove that the sum of the following subspaces is R^3? U = {(x,y,z) : x - y = z} W = {(t,-t,-t) : t∈R} I guess I need to show that any vector (x,y,z)∈R^3 can be written as the sum of a vector from U and a vector from W, but I'm not sure how to do that. I know intuitively that...
  12. J

    Subspaces of Vector Space V in R4: U = {x ∈ R4 : x1 - 2x2 - 3x3 + x4 = 0}

    Homework Statement For each of the following subsets U of the vector space V decide whether or not U is a subspace of V . Give reasons for your answers. In each case when U is a subspace, find a basis for U and state dim U Homework Equations V=P_{3} ; U=\left\{p\in\...
  13. T

    General questions about intersection of subspaces

    Hey Guys. I have some questions about vector spaces, I would really apreciate if somone could read this and let me know if I understand things or not, and if not let me know where I have it wrong. I am having a lot of trouble UNDERSTANDING how to find the intersection of two vector spaced...
  14. M

    Help w/ Subspaces Questions for Assignment Due Monday

    Subspaces Questions Help Please! Hi I have an assignment due Monday morning and there are a few questions I am not sure about or if I proved them properly: Ok so for 2b) I said that it is not a subspace because f(x)=7 when x=0, and this function never equals zero, and since this is...
  15. M

    Visualizing Subspaces and Subsets (in R3)

    I have trouble visualizing what exactly these are. Vector Space, Subset, Sub Space... What's the difference and how can I "see" it. I'm a very visual person.
  16. W

    Are These Vectors Subspaces of R3 and Do They Span the Space?

    Homework Statement 1) Determine if a) (a,b,c), where b=a+c b) (a,b,0) are subspaces of R3 and 2) Determine whether the given vectors span R3 a) v1 = (3,1,4) v2 = (2,-3,5) v3 = (5,-2,9) v4 = (1,4,-1) Homework Equations - If u and v are vectors in W, then u + v is in W -...
  17. S

    Proofs of subspaces in R^n (intersection, sums, etc.)

    Homework Statement Let E and F be two subspaces of R^n. Prove the following statements: (n means "intersection") If EnF = {0}, {u1, u2, ..., uk} is a linearly independent set of vectors of E and {v1, v2,...vk} is a linearly independent set of vectors Note: Above zero denotes the...
  18. G

    Subsets and subspaces of vector spaces

    Homework Statement T = {(1,1,1),(0,0,1)} is a subset of R^{3} but not a subspace sol i have to prove it holds for addition and scalar multiplication so let x=(1,1,1) and y =(0,0,1) so x+y = (1,1,2) so it holds let \alpha = a scalar then \alphax = (\alpha,\alpha,\alpha)...
  19. T

    Linear algebra question Subspaces

    Hey guys, new to the forum here, and its midterm time and I am working through a few questions and I can't seem to figure this one out. Homework Statement Let S = { (a,b) | b > 0 } and define addition by (a,b) + (c,d) = (a*d + a*c, b*d) and define scalar multiplication by k(a,b) = (...
  20. J

    Proving Vector Subspace: V in W iff V+W in W

    I have been given that V is a finite dimensional vector space over a field F and that W is a subspace of V. I need to show that v is an element of W if and only if v+w is an element of W. I know that because it is an 'if and only if' proof it needs to proved in both directions but don't...
  21. D

    Can vector spaces and their subspaces be visualized effectively?

    The linear algebra course I'm taking just became very "wordy" and I am having a hard time dealing notions such as subspaces without a diagram. I was thinking Venn diagrams could be used to visualize relationships between subspaces of vector spaces. Has this been a useful way to organize the...
  22. K

    Finding the Inverse of a Matrix Mapping on a Linear Subspace

    Homework Statement Let's say I'm given two vectors v_1 = \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \end{pmatrix}, v_2 = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \end{pmatrix} \in \mathbb R^4. Let W be subspace spanned by these vectors, and define G = v_1 v_1^T + v_2 v_2^T a matrix mapping \mathbb R^4 \to...
  23. K

    Inverse Matrices on Subspaces

    Homework Statement This is for a much larger question I'm working on. I have four linearly independent vectors in \mathbb C^9 , and hence they span a four dimensional space. Now I have a matrix that is composed out of their outerproduct, namely, if we let the vectors be v_i, i = 0, \ldots...
  24. M

    Common complementary vector subspaces

    Homework Statement Show that any two subspaces of the same dimension in a finite-dimensional vector space have a common complementary subspace. [You may wish to consider first the case where the subspaces have dimension 1 less than the space.] The Attempt at a Solution I've managed to sort...
  25. B

    Sums of Subspaces: Is Addition Commutative & Associative?

    If U_1, U_2, U_3, are subspaces of V (over fields R and/or C), is the addition of the subspaces commutative and associative? To me it seems rather trivial .. Since their summation is simply the set of all possible sums of the elements of U_1, U_2, U_3, and the elements themselves are...
  26. B

    Can P(F) be written as a direct sum of two subspaces?

    I'm going through Axler's book and just got introduced the concept of sums of subspaces and the direct sums. Here's one of the examples he has. Now the other examples he had were kind of trivial (such as \mathbb{R}^2 = U \oplus W where U = \{ (x,0) | x \in \mathbb{R} \} and W = \{(0,y) |...
  27. R

    Just randomly making up some subspaces

    Homework Statement Prove or give a counterexample: if U1, U2, W are subspaces of V such that: U1 + W = U2 + W then U1 = U2Homework Equations The Attempt at a Solution I would be inclined to say that it's true, however I took a peek at the back of the book and that's incorrect. Here's why I...
  28. B

    Can two subspaces have vectors in common

    Homework Statement Can two 4-dimensional subspaces of F62 have exactly 9 vectors in common? Can they have exactly 8 vectors in common? F62 is the 6-dimensional field where each (a1, a2, a3, a4, a5, a6) is an element of F2. The Attempt at a Solution F62 obviously has 26 = 64...
  29. P

    Prove the sum of two subspaces is also a subspace.

    Homework Statement Hi I'm trying to prove that the sum of two subspaces U and W is also a subspace. Homework Equations U is a subspace of V if U is also a vector space and it contains the additive identity, is closed under addition, and closed under scalar multiplication. The definition of...
  30. W

    Number of subspaces of a vector space over a finite field

    Homework Statement Prove: If V is an n-dimensional vector space of a finite field, and if 0 <= m <= n, then the number of m-dimensional subspaces of V is the same as the number of (n-m)-dimensional subspaces. The Attempt at a Solution Well here's a sketch of my argument. Let U be an...
  31. E

    Bounding the p-norms on l_p sequence subspaces

    Homework Statement Let 1\leq r<\infty and x\in\ell_{r}=\left\{ x \text{ is a sequence with } \sum_{n=1}^{\infty}\left\vert x_{n}\right\vert^{r} \text{ converges.}\right\}, then \left\vert\left\vert x\right\vert\right\vert_{\infty}=\lim_{r\rightarrow\infty}\left\vert\left\vert...
  32. P

    Determine whether the following subsets are subspaces

    Homework Statement H = {(x,y,z) \in R^3 | x + y^2 + z = 0} \subseteq R^3 T = {A \in M2,2 | AT = A} \subseteq M2,2 The Attempt at a Solution Our lecturer wasn't quite clear about how to go about this. He talked out closed under addition and multiplication but that's about it...
  33. D

    Is the Set of Solutions to a Homogeneous System of Equations a Subspace?

    Homework Statement Okay, this is the last True/False question I will post. True or False: \text{The set of all solutions to the }m\times n\text{ homogeneous system of equations }Ax=0\text{ is a subspace of }\mathbb{R}^m. Homework Equations None The Attempt at a Solution I...
  34. Saladsamurai

    Prove that the intersection of subspaces is subspace

    Homework Statement Prove that the intersection of any collection of subspaces of V is a subspace of V. Okay, so I had to look up on wiki what an intersection is. To my understanding, it is basically the 'place' where sets or spaces 'overlap.' I am not sure how to construct the problem...
  35. A

    Is the Solution Set of Ax=b a Subspace of R^n?

    Please anyone solve this question or can even email me on my ID abu_95bakar@yahoo.com... For the following question determine whether the set S is a sub space of the given vectorspace V. v=Rn( where n represent dimension), S is the solution set of the sysytem Ax=b, where A is an mxn...
  36. Saladsamurai

    Determine whether Subsets are Subspaces

    Here we go...wheeeee Homework Statement For each of the following subsets of F3, determine whether it is a subspace of F3 (a) {(x_1, x_2, x_3) \in \mathbf{F}^3: x_1+2x_2+3x_3=0} (b) {(x_1, x_2, x_3) \in \mathbf{F}^3: x_1+2x_2+3x_3=4} (c) {(x_1, x_2, x_3) \in \mathbf{F}^3: x_1x_2x_3=0} (d)...
  37. Saladsamurai

    Sum of Two Subspaces: Exploring the Definition

    This is an example that I am a little confused by: U={(x,0,0)\in\mathbf{F}^3:x\in\mathbf{F}}\text{ and }W={(0,y,0)\in\mathbf{F}^3:y\in\mathbf{F}} Then U+W={(x,y,0):x,y\in\mathbf{F} Okay, I get that. Now it says that U is defined the same as above but now let...
  38. D

    Calculating Linear Span: Vector a1 (-7, 8, 5) and Line Equation

    Homework Statement (i)Show that the linear span of the vector a1 = (-7, 8, 5) is the line whose equation is x/(-7) = y/8 = z/5 The Attempt at a Solution The problem is, I don't know where or how to start.
  39. B

    Subspace & Dimension of V in P2 - Closure and Dimension Analysis

    Homework Statement Is the collection a subspace of the given vector space? If so what is the dimension? V={ax^2+bx+c: a=b+c} in P2 Homework Equations The Attempt at a Solution The first part of the question is pretty straightforward. I just verified closure under addition and...
  40. Deneb Cyg

    Linear transformations and subspaces

    Homework Statement Let B={b1,b2} be a basis for R2 and let T be the linear transformation R2 to R2 such that T(b1)=2b1+b2 and T(b2)=b2. Find the matrix of T relative to the basis B. The Attempt at a Solution I know that the matrix I'm looking for needs to be 2x2 and that the standard matrix...
  41. L

    Counting 1-D Subspaces of Z_3^3

    how many 1 dimensional subspaces of Z_3^3 are there? Z_3^3 has 3^3 = 27 vectors 26 of which are non zero then we can say v and 2v have the same span and so there are in fact 13 1 dimensional subspaces. is this true?
  42. T

    Determining whether or not (a,0,0) and (a,b,0) are subspaces of R3

    Homework Statement "use theorem (below) to determine which of the following are subspaces of R3: (a,0,0) and (a,b,0) Homework Equations The theorem: W is a subspace of V iff: - u and v are vectors in W, u + v is in W - k is a scalar, u is a vector in W, then ku is in W...
  43. 0

    Linear Algebra: Vector Spaces, Subspaces, etc.

    Homework Statement Which of the following subsets of R3? The set of all vectors of the form a) (a, b, c), where a=c=0 b) (a, b, c), where a=-c c) (a, b, c), where b=2a+1Homework Equations A real vector space is a set of elements V together with two operations + and * satisfying the following...
  44. D

    Union of subspaces of a linear space

    Is there a linear space V in which the union of any subspaces of V is a subspace except the trivial subspaces V and {0}? pls help
  45. K

    I still don't quite get the idea of subspaces, span, and range

    Alright. As stated in the title above. So, a subspace is a set of vectors that satisfies: 1) It contained the zero vector; 2) It's closed under addition and subtraction. By "closed", it means that when I add another vector in R2 or multiply by a scalar k on A(x)=m, it will end up with...
  46. S

    Finding invariant subspaces

    Homework Statement Let V be a finite dimensional, nonzero complex vector space. Let T be be a linear map on V. Show that V contains invariant subspaces of dimension j for j=1, ..., dim V. Homework Equations Since V is complex, V contains an invariant subspace of dimension 1. The...
  47. C

    Vector Subspaces, don't understand

    Vector Subspaces, don't understand... Homework Statement Which of the given subsets of the vector space, M23, of all 2 X 3 matrices are subspaces. (a) [a b c, d 0 0] where b = a + c Homework Equations Theorem 4.3 Let V be a vector space with operations + and * and let W be a...
  48. C

    Sums of Subspaces: U+U, U+V, Is U+W=W+U?

    Homework Statement Let U and W be subspaces of V. What are U+U, U+V? Is U+W=W+U? Homework Equations The Attempt at a Solution It is easy to show that U+U and U+V are spaces too under closed addition and scalar multiplication, but I'm not sure where they lie. For example, is U+U a...
  49. W

    Linear Algebra: Vector Subspaces

    Homework Statement True/false: Union of two vector subspaces is a subspace. Homework Equations none The Attempt at a Solution I'm unsure if this is true because I'm also unsure if it already assumes that it is closed under scalar multiplication and addition. If it is closed, then...
  50. G

    Subspaces in Polynomial P_5(x) of Degree < 5

    If P_{5}(x) is the set of all polynomials in x in degree less than 5. Which of following subsets of P_{5}(x) are subspaces. (i) the set of all polynomials in P_{5}(x) of even degree (ii) the set of all polynomials in P_{5}(x) of degree 3 (iii) the set of all polynomials p(x) in P_{5}(x) such...
Back
Top