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. A

    Understanding Vector Spaces and Subspaces

    hi, I am confused about vector spaces and subspaces. I've just started a book on linear algebra, and i understood the 1st chapter which delt with gaussian reduction of systems of linear equations, and expressing the solution set as matricies, but the 2nd chapter deals with vectors and I'm...
  2. V

    Subspaces of V: F(R, R)

    V = F(R, R), the vector space of all real valued functions f(x) of a real variable x. Which are subspaces of V? (A) {f | f(0) = 0} (B) {f | f(0) = 1} (C) {f | f(0) = f(1)} (D) C^0(R) = {f | f is continous} (E) C^1(R) = {f | f is differentiable and f' is continous}...
  3. V

    LINEAR ALGEBRA: Consider 2X2 Matrices - What are the subspaces?

    Consider 2-by-2 matrices \mathbf{A} =\left( \begin{array}{cc}a & b \\c & d \\\end{array} \right) \in \mathbbm{R}^{2 X 2}. Which of the following are subspaces of \mathbbm{R}^{2 X 2}? (A) {A | c = 0} (B) {A | a + d = 0} (C) {A | ad - bc = 0} (D) {A | b = c} (E) {A | Av = 2v}, where...
  4. V

    Algorithms for quantifying intersections of subspaces

    Greetings, I'd like to know how one goes about finding a basis for the intersection of two subspaces V and W of a given vector space U. I am aware of the identity V \cap W = (V^{\per} \cup W^{\per})^{\per} (essentially the orthogonal space of the union of orthogonal spaces of V and W), but this...
  5. JasonJo

    Solving Vector Space and Subspace Problems in Linear Algebra

    i've been having some trouble with my linear algebra homework and I am wondering if you guys could give me some insight or tips on these problems: Let v be any vector from V, and let a be any real number such that av=0. Show that either a=0 or v=0. - i was thinking about assuming the...
  6. P

    Equal Subspace A & B: Is A=B?

    A and B are two subspaces contained in a finite vector space V and dimA = dimB Can we conclude A=B? In that subspaces A and B are really the same subspace and every element in one is in the other? I think yes because if dimA=dimB then their basis will contain the same number of vectors...
  7. MathematicalPhysicist

    Cyclic Subspaces: Proving Equality of Zero Spaces for Coprime Polynomials

    prove that Z(v,T)=Z(u,T) iff g(T)(u)=v, where g(t) is prime compared to a nullify -T of u. (which means f(t) is the minimal polynomial of u, i.e f(T)(u)=0). (i think that when they mean 'is prime compared to' that f(t)=ag(t) for some 'a' scalar). i tried proving this way: suppose, g(T)(u)=v...
  8. T

    Finding the Intersection of Subspaces with Given Spanning Vectors

    How do you find the intersection of subspaces when the subspaces are given by the span of 3 vectors? For example, U is spanned by { X1 , X2 , X3} and V is spanned by { Y1, Y2, Y3}. Thanks in advance.
  9. M

    Infinitely many subspaces in R3 ?

    infinitely many "subspaces" in R3 ? In R3, there are zero, 1, 2, 3 dimensional subspaces. But how can I express them with 'specific' example, using variables x,y,and z?
  10. Pengwuino

    Subspaces in R4: Get Started & Understand Now

    I'm so lost! 1. W is the set of all vectors in R4 such that x1 + x3 = x2 + x4. Is W a subspace of R4 and Why? How do i get started here? I'm thoroughly confused on this whole idea of vector spaces and such.
  11. T

    Does the Addition of Subspaces Always Result in Unique Additive Identities?

    Suppose U is a subspace of V. Then U+U = U+{0}=U, right? So the operation of addition of vector spaces does not have unique additive identities. *typo in title
  12. mattmns

    Linear Algebra - Subspaces proof

    Hello, just wondering if my proof is sufficient. Here is the question from my book: Show that the following sets of elements in R2 form subspaces: (a) The set of all (x,y) such that x = y. ------- So if we call this set W, then we must show the following: (i) 0 \in W (ii) if v,w \in W, then...
  13. B

    Subspaces and inner products

    Hi, can someone shed some light on the following question. It's been bothering me for a while and I'd like to know where I went wrong. Here is what I can remember of the question. The following is an inner product for polynomials in P_3(degree <= 3): \left\langle {f,g} \right\rangle =...
  14. E

    Two quick questions on subspaces (intro to tensor calc)

    So the parametric equations of a hypersurface in VN are x^1=acos(u^1) x^2=asin(u^1)cos(u^2) x^3=asin(u^1)sin(u^2)cos(u^3) ... x^(N-1)=asin(u^1)sin(u^2)sin(u^3)...sin(u^(N-2))cos(u^(N-1)) x^N=asin(u^1)sin(u^2)sin(u^3)...sin(u^(N-2))sin(u^(N-1)) where a is a constant. How do I find the...
  15. B

    Subspaces, R^n How to visualize?

    Hey, I have no problems dealing with vectors in space, R^3. But I am having a lot of trouble with vectors in R^n. One of my basic questions is what is R^n. I mean doesn't the vector space already encompass everything? How do I visualize R^n vectors? Can you reccomend any good online tutorials...
  16. C

    One more ? proving subspaces

    see attached equation thank you!
  17. B

    Exploring Subspaces and Dimension in Z_2^3

    Hi, I'm wondering how I would decide how many "subspaces of each dimension Z_2^3 has." The answer is: 1 subspace with dim = 0, 7 with dim = 1, 7 with dim = 2, 1 with dim = 3. I'm looking for subsets of Z_2^3 which are closed under addition and scalar multiplication. An arbitrary vector in...
  18. R

    Vector Spaces, Subspaces, Bases etc

    Vector Spaces, Subspaces, Bases etc... :( Hello. I was doing some homework questions out of the textbook and i came across a question which is difficult to understand, could somebody please help me out with it? -- if U and W are subspaces of V, define their intersection U ∩ W as follows...
  19. R

    Vector Space & Vector Subspaces

    Hello... I've been doing some home work on Vector Spaces and Vector Subspaces and I need help solving a problem... Can somebody please help me? Consider the differential equation f'' + 5f' + 6f' = 0 Show that the set of all solutions of this equation is a vector subspace of the...
  20. G

    Are Continuous Functions with Zero Integral a Subspace of C[a,b]?

    Yet another problem I need to get some starting help on: Show that the set of continuous functions f=f(x) on [a,b] such that \int \limits_a^b f(x) dx=0 is a subspace of C[a,b] Thank you
  21. B

    Proving the Intersection of Subspaces is a Subspace: A Vector Space Proof

    Hi can someone please help me with the following question. Such questions always trouble me because I don't know where to start and/or cannot continue after starting. Q. Let H and K be subspaces of a vector space V. Prove that the intersection of K and H is a subspace of V. By the way...
  22. L

    What Is the Intersection of Subspaces U and V in R^3?

    I have 2 subspaces U and V of R^3 which U = {(a1, a2, a3) in R^3: a1 = 3(a2) and a3 = -a2} V = {(a1, a2, a3) in R^3: a1 - 4(a2) - a3 = 0} I used the information in U and substituted it into the equation in V and I got 0 = 0. So, does it mean that the intersection of U and V is the whole...
  23. K

    U a Subspace of R3: Yes- Explained Here

    Q: Determine whether U is a subspace of R^3. U = {[0 s t]^T | s and t in R} A: Yes. U = span {[0 1 0]^T, [0 0 1]} Can someone explain to me how the heck they come up with that answer? Seems so random.
  24. H

    Separation axioms vs subspaces

    How do separation axioms carry over to subspaces? Some are clear -- it's easy to see that if any two points of a space X are separated by neighborhoods, then the same must be true of any subset S of X. But what about the nicer ones? Is it true that if S is a subset of a normal space, that...
  25. G

    Existence of n T-invariant Subspaces in Vector Spaces and Linear Operators

    Hi everyone- any help would be great! For each integer n>= 2, there exists a vector space V and a linear operator T : L(V ) such that V has exactly n T- invariant subspaces. I think it is true but i do not know how to prove it... awesome thanks!
  26. Z

    Understanding Metric Spaces: Subsets vs. Subspaces

    Having some difficult with general concepts of metric spaces: 1) What is the difference between a subset and a subspace. let's say we have metric space X. and A is a set in that space. Is A necessarily a metric space itself? 2) Why is the metric of X ( d(x,y) for x,y belonging to X )...
  27. M

    What are the properties of subspaces?

    I had a question regarding subspaces. Given vectors (a,b,c,d) s.t. \left{\mid}\begin{array}{cc}a&b\\c&d\end{array}\right{\mid}=0 a supspace of \Re^4? Though i kno the answer is yes, but i don't understand like it looks to me that it uses one of the three properties of subspaces; the zero...
  28. G

    Does Every Linear Operator Have a Nontrivial Invariant Subspace?

    Does every linear operator have a nontrivial invariant subspace? My professor mentioned this question in class, but never actually answered it. I am curious if this is true or not and why.
  29. G

    Proving U1=U2 When U1, U2, W are Subspaces of V

    I have to prove or give a counter example to the statement if U1, U2, W are subspaces of V such that V=U1 direct sum W and V=U2 direct sum W, then U2=U1. This is what I did: Let v be an element of V. Then v=v1+v2 for v1 an element of U1 and v2 and element of W and v=v3+v2 for v3 an...
  30. P

    Intersecting subspaces in N dimensions

    Hey guys, I have a little problem here: given two subspaces U and W both of dimension two of an N dimensional space show in general that if N = 3 the intersection of U and W forms a curve; if N = 4 a finite number of points; and N > 4 they do not in general intersect at all. I can kind...
  31. P

    Intersecting Subspaces in N dimensions

    Hey guys, I have a little problem here: given two subspaces U and W both of dimension two of an N dimensional space show in general that if N = 3 the intersection of U and W forms a curve; if N = 4 a finite number of points; and N > 4 they do not in general intersect at all. I can kind...
  32. F

    On proving real vector spaces (subspaces)

    I hope someone can help me (guide) in this theorem. How can I show that a "subset W of a vector space V is indeed a subspace of V if and only if given u and v as vectors in W and a and b are said to be scalars, then au + bv is in W."? Can I assume a vector with my desired number of...
  33. S

    Can't solve a problem about subspaces

    I can't solve a problem about subspaces. Help would be great! U and V are subspaces in the vector space R^4[x] given with: U={p(x)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4; a1+a2+a3+a4=0, a1+a2+2a3+2a4=0, a0+a1=a3+a4} V=L{x^3-x^2+x, x^4+1} Find the dimensions and basis for U, U+V and U?V. Is...
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