Subspace Definition and 574 Threads

Homework Statement Is the set of all vectors in R^n whose components form an arithmetic progression a linear subspace of R^n? Homework Equations none The Attempt at a Solution I basically need one thing verified: would (0,0,0,...,0) be considered an arithmetic progression. The...

I'm trying to show that a set W of polynomials in P2 such that p(1)=0 is a subspace of P2. Then find a basis for W and dim(W). I have already found that the set W is a subspace of P2 because it is closed under addition and scalar multiplication and have showed that. The thing I'm stuck on...

Trying to solve a question in linear algebra. P2 is a polynomial space with degree 2. Is P(t): P'(1)=P(2) (P' is the derivative) a subspace of P2. What is the basis ? It seems that it is a subspace with basis 1-t,2-t2. Can anybody explain how this can be found?

Let A_i (i=1,...,k) be a nonsingular complex matrix which size is M by M. The question is how to find a complex matrix X which size is M by N such that: span(A_1*X)=...=span(A_k*X) (I guess that there must be relations between M,N and k when nontrival solution exists. ) ask: 1)if non...

Homework Statement Find a basis for F=\left\{(x,y,z,w): -x+y+2z-w=0\right\}The Attempt at a Solution So this looks like a plane to me, but I find 4-d space confusing, so that might be wrong. It does have the form \mathbf{x}^T\mathbf{n}=0, so that's kind of where I'm getting the idea that it's...

Homework Statement let V be the vector space consisting of all infinite real sequences. Show that the subset W consisting of all such sequences with only finitely many non-0 entities is a subspace of VThe Attempt at a Solution Ok so i have to show 1.Closure under Addition,2. Closure under...

Homework Statement Let V be a finite dimensional subspace. Let V\supseteqU1\supseteqU2\supseteq...\supseteqUk. Show that there exists k such that Uk=Uk+1=...=Un=...Homework Equations We were also told to assume none of the subspaces are zero dimensional, and to think about how the dimensions...

Homework Statement Find a basis for each of these subspaces of R4 All vectors that are perpendicular to (1,1,0,0) and (1,0,1,1) 2. The attempt at a solution I'm not sure how to approach this question. The only thing I can think of is that a vector that would be perpendicular to both would be...

Homework Statement Suppose A is an unbounded subspace of a metric space (X,d) (where d is the metric on X). Fix a point b in A let B(b,k)={a in X s.t d(b,a)<k where k>0 is a natural number}. Let A^B(b,k) denote the intersection of the subspace A with the set B(b,k). Then the...

Homework Statement The Attempt at a Solution The terminology in this question confuses me into what I am actually trying to solve. It seems to me that S-perp would naturally be a subspace of real column vectors based on the fact that we specify that S\neq0. It goes on to mention...

Homework Statement Determine whether or not the given set is a subspace of the indicated vector space: Functions f such that [integral from a to b]f(x)dx = 0; C[a,b] (not sure how to do the coding for integrals) Homework Equations to be a subspace it must follow these axioms: (i) if x and y...

Hey guys, I've got another problem I could use some assistance with. "In this problem we suppose that F is a field, A is an m by n matrix over F and that W is a subspace of Fm. (a) Show that U = {v \in Fn: Av \in W} is a subspace of Fn. (b) Now suppose that m = n and A is invertible, and that B...

Hey guys. I came upon this problem in the professor's recommended problems, and I have no idea how to solve it. After spending an hour on it, I've got nothing. Any suggestions would be much appreciated! A function f : R -> R is called even just in case f(-r) = f(r) for every real number r. f...

Suppose U and V are finite-dimensional subspaces of some finite-dimensional space W (over the field Q in my actual case, but probably irrelevant) I've got a subspace in mind that I can't quite define well. I'll start with a concrete example: Consider R^3 with orthonormal basis {x, y, z}. Let U...

Given V = {(x1; x2;….; xn) | Σni=1 xi=0} (sum of vectors is equal to zero) be a subspace of Rn. How can we find a basis of V such that for each vector {(x1; x2;….; xn) in the basis Σi=1n x2i=1 ( i.e. sum of squares is equal to 1).

Let X be a normed vector space. If C is a closed subspace x is a point in X not in C, show that the set C+Fx is closed. (F is the underlying field of the vector space).

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...

Homework Statement Hi I need help understanding a proof. This is my first time in a pure math class, so proofs of this type are a little weird to me. If U is a subspace of the vectorspace V, what is U+U? Homework Equations The proof: (v_{1}+v_{2})\in U+U As v_{1},v_{2}\in...

Homework Statement For the following subset W of R3 determine whether or not W is a subspace of R3. If the subset is not a subspace give a specific example to indicate why it is not a subspace. ii.) W = {(x,y,z): 2x + y + 3z = 0 The Attempt at a Solution I know how to do this...

Homework Statement Let V and W be vector spaces over F and T:V \rightarrow W a linear transformation. Prove that ker(T):={\epsilon V\mid T()=0_{v}} is a vector subspace of V Homework Equations The Attempt at a Solution Is it all right just to state the trivial solution. ie...

Hi, I wonder if there is some agreed-upon best way to reconstruct the matrix of a positive definite operator A using "sampling" (like in tomography). More in detail I want to do this: I have many small sets of basis functions. The sets are in general not orthogonal. I compute matrix elements...

There's a problem from Rudin's Functional Analysis where I need to show something is a dense subspace of 1st category. But I thought that it was the definition of dense that its closure is the whole space. Hence the closure doesn't have empty interior. So the dense subspace can't be 1st...

Hi. if anybody can help me Let S be the matrix a b c d with a constrain a = -2d, b= 3c -d Prove that S is a subspace of R2?

Homework Statement Let W1 and W2 be two subspaces of R^n. Prove that their intersection is also a subspace. Homework Equations The Attempt at a Solution I know that in R^2 and R^3 the intersection would be the origin, which would be the zero vector, which would be a 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...

Here we go again:biggrin: Give an Example of a subset U of R^2 that is closed under scalar multiplication, but is not a subspace of R^2 I am thinking let U={(x1,x2) : x1=-x2} If x=(x1,x2) and y=(y1, y2) then x+y= (-x2-y2), x2+y2) = (-(x2+y2), x2+y2) okaaayy so that does not work...

Homework Statement Give an Example of a subset U of R2 that is closed under addition and under taking additive inverses (i.e., -u in U whenever u in U), but is not a subspace of R2 Okay, I know that this problem is not hard, but I just need a hint. I don't want to just start arbitrarily...

Okay then. I just read the section of Axler on subspaces. It says that if U is a subset of V, then to check that U is a subspace of V we need only check that U satisfies the following: additive identity 0\in U closed under addition u,v\in U\text{ implies }u+v\,\in\,U closed under scalar...

Homework Statement From Introduction to Topology by Bert Mendelson, Chapter 2.7, Exercise 8: Consider the subspace (Q, d_Q) (the rational numbers) of (R, d). Let a1, a2, ... be a sequence of rational numbers such that \lim_{n} a_n = \sqrt{2}. Does the sequence converge when considered...

Homework Statement Find an example of subspaces W1 and W2 in R^3 with dimensions m and n, where m>n>0, such that dim(intersection of W1 and W2)= n Homework Equations dim(W1+W2)= dim(W1) + dim(W2)-dim(intersection of W1 and W2) The Attempt at a Solution Well what I know...

Homework Statement Suppose T is a linear operator on R^4 such that T(a,b,c,d) = (a + b, b - c, a + c, a + d). Find a basis for the T-cyclic subspace of R^4 generated by z = (1, 0, 0, 0) Homework Equations The Attempt at a Solution I found a basis, but I don't think the method I used was the...

Homework Statement Let S denote the collection of all polynomials of the form p(t) = (2a - b)t^2 + 3(c - b)t + (a - c), where a,b,c are real numbers. Determine whether or not S is a subspace of P2. The Attempt at a Solution Okay, so I know that in order for S to be a subspace, it must...

Homework Statement Determine whether or not the set of all functions f such that f(1)+f(-1)=f(5) is a subspace of the vector space F of all functions mapping R into R. Homework Equations The Attempt at a Solution I think that (f(1)+f(-1))+(g(1)+g(-1))=(f+g)(1)+(f+g)(-1)=(f+g)(5)...

Homework Statement determine if the space is a subspace testing both closure axioms. in R^2 the set of vectors (a,b) where ab=0 Homework Equations The Attempt at a Solution i just used the sum and product which are the closure axioms. But at the end how do you tell if the...

Homework Statement Determine if the sets are a subspace of the real vector space: Prof is kinda hard to hear and doesn't explain stuff that well, can I get some help with this one? Homework Equations H = {[a,b,c,d] exist in 4-space| 4a+2b-8c+2d = 3a-5b+6d = b-6c-2d = 0} H =...

If a problem I'm doing asks to find V2 where V is a vector is it simply the dot product of the vector, or the cross product? The question: Which of the following sets of vectors v = {v1,...,vn} in Rn are subspaces of Rn (n>=3) iii) All v such that V2=V12 He proved it by saying...

Hey, this isn't for homework per se, but if anyone could lend me a hand figuring this out I'd appreciate it a lot! Homework Statement Determine whether Q is a subspace of R2/R3 in the following cases: Homework Equations Q = \{\left v = \left( v1, v2, 0 \right) | v1,v2 \in R...

Homework Statement Let W be the subspace spanned by the given column vectors. Find a basis for W perp. w1= [2 -1 6 3] w2 = [-1 2 -3 -2] w3 = [2 5 6 1] (these should actually be written as column vectors. Homework Equations The Attempt at a Solution So, I...

Hi, I would like to know why the set of all n*n matrix whose determinant is zero is not a subspace of Mn,n .Can anyone explain the reason for me? Thanks!

Homework Statement Prove that the set of all 3-vectors orthogonal to [1, -1, 4] forms a subspace of R^3. Homework Equations Orthogonal means dot product is 0. The Attempt at a Solution I know the vectors in this subspace are of the form [a,b,c] where a - b + 4c = 0. However I...

Homework Statement I am trying to solve this problem: Let W_1, W_2, W_3 be subspaces of a vector space, V. Prove that W_1 ∩ (W_2 + ( W_1 ∩ W_3)) = (W_1 ∩ W_2) + (W_1 ∩ W_3). Can someone help me show this? I have tried using Dedekind's law, but not sure it that is the way to go. The...

Homework Statement Find a projection [matrix] E which projects R2 onto the subspace spanned by (1,-1) along the subspace spanned by (1,2).Homework Equations P = \frac{a a^{T}}{a^{T} a}The Attempt at a Solution Computing P... P = \[ \left( \begin{array}{ccc} \frac{1}{2} & -\frac{1}{2}\\...

I have a problem. Suppose that {u1,u1,...,um} are vectors in R^n. Prove, dircetly that span{u1,u2,...,um} is a subspace of R^n. How would I go by doing this?

Homework Statement Let U={(x,y,z) \in R3 : x=z}. Show that U is a subspace of R3. Homework Equations The Attempt at a Solution U is non-empty it contains the 0 vector: U= {(x,y,z) = (s,t,s), s,t \in R} U={s(1,0,1)+t(0,1,0), s,t \in R} for s,t=0...

I saw this problem in a book, it asks if there are two subspaces of Rn, say U & V and the following condition is true: W={w \in R^n : w=u+v for some u \in U and v \in V} Make a proof/show that W is a subspace of Rn. I think maybe we need to try to somehow prove that the set W is a...

Homework Statement find a basis for the subspace R^5 that consists of all the vectors of the form [(b-c), (d-2b), (4d), (c-2d), (6d+2b)] Homework Equations The Attempt at a Solution the only solution I can think of is e1, e2, e3, e4, e5... I don't think it's that simple...

Homework Statement find a basis of the subspace W:=A\in M2*2(R) : trace (A)=0 of the vector space M2*2 (R) and hence determine the dimension of W Homework Equations The Attempt at a Solution trace(A) denote the the sum of the diagonal elements of the matrix A=aij do i need to...

Homework Statement 1) Explain why the set W={(x,y)inside dimension 2; |x|=|y|} is not a real subspace 2) Show that the set V={[a b];a+d=0} is a real subspace of dimension 3 {[c d] } Homework Equations The Attempt at a Solution

Homework Statement Say you have the plane given by equation 4x + 3y + 4z + 4 = 0 This plane is not a subspace of R^3, right? My reasoning is that this plane can't include the origin, but I just need some clarification to make sure that I understand what a subspace is. Thanks...

It is a fact that if X is a compact topoloical space then a closed subspace of X is compact. Is an open subspace G of X also compact? please consider the following and note if i am wrong; proof: Since G is open then the relative topology on G is class {H_i}of open subset of X such that the...