Subspaces Definition and 317 Threads

Suppose I have a nondegenerate alternating bilinear form <,> on a vector space V. Under what conditions would a subspace U of V retain nondegeneracy? That is, if u ∈ U and u ≠ 0, then could I find a w ∈ U such that <u,w> ≠ 0? So for example, it's clear that no one-dimensional subspace W of V...

Hi, I was wondering if someone could check my work for this linear algebra problem. I have attached the problem statement in the file "problem" and my work in the file "work." I would type out my work on here, but I couldn't figure out how to put matrices in a post so I just took a pic of my...

Homework Statement Let E = {“ax+by+cz = d” | a; b; c; d ∈ R} be the set of linear equations with real coefficients in the variables x, y and z. Equip E with the usual operations on equations that you learned in high school. addition of equations, denoted below by “⊕” and multiplication by...

Homework Statement Which of the following are subspaces of F[R] = {f |f:R-->R}? a) U = {f e F[R]|f(-1)f(1)=0 b) V = " |f(1)+f(2)=0 c) S = " |f(x)=f(-x) d) T = " |f(1)<= 0 Homework Equations The Attempt at a Solution I got S and V or c) and b), is that correct? I...

When studying linear algebra when encountering a system Ax=b, I always read of the fundamental subspaces of A: N (the null space, all solutions x of Ax=0), the column or domain space of A: (the space spanned by the columns of A, or in other words, all possible b for Ax=b), the row space (the...

Homework Statement Determine whether the following are subspaces of P4: a) The set of polynomials in P4 of even degree b) The set of all polynomials of degree 3 c) The set of all polynomials p(x) in P4 such that p(0) = 0 d) The set of all polynomials in P4 having at least one real root The...

Could someone please help me with the following question with a guided step by step answer: Show that T = (x, y, z) : -1 ≤ x + y + z ≤ 1 is not a vector subspace of R3 Thanks!

Homework Statement Let W1 be the set of all nxn skew-symmetric matrices with entries from a field F. Assume F is not characteristic 2 and let W2 be a subspace of Mnxn(F) consisting of all nxn symmetric matrices. Prove the direct sum of W1 and W2 is Mnxn(F). Homework Equations The...

Hi: I have a problem about combine bases from subspaces. This is part of orthogonality. The examples as following: For A=##\begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix}## split x= ##\begin{bmatrix} 4 \\ 3 \end{bmatrix}## into ##x_r##+##x_n##=##\begin{bmatrix} 2 \\ 4...

Homework Statement -Problem number 1 Given the set {u ,v} , where u=(1,2,1) and v=(0,-1,3) in R^3 find an equation for the space generated by this set. -Problem number 2 The subspace S is defined as S= {(x,y,z) : x + 2y - z =0} find a set B={u,v} in R^3 such that each...

Homework Statement If (V,\omega) is a symplectic vector space and Y is a linear subspace with \dim Y = \frac12 \dim V show that Y is Lagrangian; that is, show that Y = Y^\omega where Y^\omega is the symplectic complement. The Attempt at a Solution This is driving me crazy since I...

Hi: I am a newbie to linear algebra; I have a problem about vector space and subspaces. How to distinguish these two subject. what I know from books is subspace is going through zero, but I still can not figure out what is the difference between vector space and subspaces, thanks.

Homework Statement Let A be a fixed 2x2 matrix. Prove that the set W = {X : XA = AX} is a subspace of M2,2. Homework Equations Theorem: Test for a subspace If W is a nonempty subset of a vector space V, then W is a subspace of V if and only if the following closure conditions hold. 1...

I have a question concerning subspaces of infinite dimensional vector spaces. Specifically given any infinite dimensional vector space V, how might one construct an infinite decreasing chain of subspaces? That is: V=V0\supseteqV1\supseteq... , where each Vi is properly contained in Vi-1...

Hi! I have the following statements in a script on Riemannian submersions: (\pi is the submersion \mathbb S^{2n+1} \rightarrow \mathbb{CP}^n or \mathbb S^{4n+3} \rightarrow \mathbb{HP}^n respectively.) Regarding a) it is then said: "Let w \in T\mathbb{CP}^n, \lambda \in \mathbb C. Let...

Hi, I have read my notes and understand the theory, but I am having trouble understanding the following questions which are already solved (I am giving the answers as well). The first question says: Let U_{1} and U_{2} be subspaces of a vector space V. Give an example (say in V=\Re^{2}) to...

Homework Statement Is the subset of P= {a0 + a1x + a2x2 + ... + anxn} formed only by the polynomials that satisfy the condition: a1a3≤0 a vector subspace? Homework Equations I already proved the subset is not closed under addition so I know it's not a vector subspace, however, the...

Homework Statement Consider the ordinary vectors in three dimensions (ax, ay, az) with complex components. a) Does the subset of all vectors with az = 0 constitute a vector space? If so, what is its dimension; if not; why not? b) What about the subset of all vectors whose z component is 1...

Homework Statement consider R^4. Let V be the set of vectors in the form ( 2x+3y, x, 0 , -x+2x) is this a subspace of R^4 and why? find a basis if it's possible Homework Equations The Attempt at a Solution I know that the set must work under scalar multiplication and vector...

Homework Statement I'm working on a problem that involves looking at the dimension of the intersection of two subspaces of a vector space. Homework Equations M \subset V N \subset V dim(M \cap N) [\vec{v}]_{B_M} is the coordinate representation of a vector v with respect to the...

I'm reading about symmetries in QM in "Geometry of quantum theory" by Varadarajan. In one of the proofs, he refers to theorem 2.1, which is stated without proof. He says that the theorem is proved in "Linear algebra and projective geometry" by Baer. That isn't very helpful, since he doesn't even...

Homework Statement Prove that C(AB) is a subset of C(A) for matrices A,B, where C denotes column space. Homework Equations C(AB) = {b \in \mathbbcode{R}^m: Ax=b is consistent} The Attempt at a Solution I don't really know where to start.

Homework Statement The Attempt at a Solution Let (x,y,z) be arbitrary. We write, (x,y,z) = a(1,0,1) + b(0,1,0) + c(0,1,1) for a,b,c \in R . From this, (x,y,z) = (a,0,a) + (0,b,0) + (0,c,c) = (a,b+c,a+c). However, (a,b+c,a+c) can generate all of R^3 for appropriately chosen a,b,c...

Homework Statement http://img824.imageshack.us/img824/3849/screenshot20120122at124.png The Attempt at a Solution Let S = \left\{ S_1,...,S_n \right\} . If L(S) = V, then T = \left\{ 0 \right\} and we are done because S + T = V. Suppose that L(S) ≠ V. Let B_1 \in T such that B_1 \notin...

Homework Statement http://img856.imageshack.us/img856/5586/screenshot20120121at328.png The Attempt at a Solution I propose the vectors x,x^2,x^3 form a basis of V. To test for linear independence, let 0 = a_1 x + a_2 x^2 + a_3 x^3, where a \in R. A polynomial is 0 iff all of its...

Homework Statement http://img21.imageshack.us/img21/4580/screenshot20120117at218.png The Attempt at a Solutiona) Suppose we have two arbitrary vectors of E, call them X,Y. Let X = (2x,x) where x is in R and let Y = (2y,y) where y is in R. If we add X and Y we have (2x,x) + (2y,y) =...

Homework Statement Hi, everybody! I'd like to ask you about the direct sum of subspaces... I refer to two linear algebra books; 1)Friedberg's book, 2)Hoffman's book. First of all, I write two definitions of direct sum of subspaces... in the book 1), Def.1). Let...

Homework Statement http://img857.imageshack.us/img857/548/screenshot20120112at853.png The Attempt at a SolutionI reasoned that if U is a vector subspace, then the zero vector must certainly be an element of U. That is, (0,0,0) \in U. If this is true, then we can write for x_1 + x_2 + x_3...

Homework Statement I'm having a tough time figuring out just how to get the orthogonal complement of a space. The provlem gives two separate spaces: 1) span{(1,0,i,1),(0,1,1,-i)}, 2) All constant functions in V over the interval [a,b] Homework Equations I know that for a subspace W of an...

Homework Statement V is a vector space with dimension n, U and W are two subspaces with dimension k and l. prove that if k+l > n then U \cap W has dimension > 0 Homework Equations Grassmann's formula dim(U+W) = dim(U) + dim(W) - dim(U \cap W)The Attempt at a Solution Suppose k+l >n. Suppose...

Homework Statement U={(x1,x2,x3)\inℝ3 | x1+x2=0} Is this a linear subspace of ℝ3?Homework Equations x1+x2=0The Attempt at a Solution I know that in order to be a linear subspace, it must be closed under addition and scalar multiplication. I'm just not really sure how to incorporate the x1+x2=0...

Homework Statement Homework Equations can someone help me to solve this problem? The Attempt at a Solution I couldn't even approach

Homework Statement Determine whether the following sets form subspaces of ℝ2: (a) {(x1, x2)T | x1 + x2 = 0} (b) {(x1, x2)T | x1 * x2 = 0}Homework Equations The Attempt at a Solution I know that a is a subspace and b is not, but I would like to know why. For part A, I let x=[c, -c]T ∂[c,-c]=...

Homework Statement See attachment The Attempt at a Solution How should I approach these questions? By using the projection formula?

Homework Statement Show that the set W consisting of all vectors in R4 that are orthogonal to both X and Y is a subspace of R4. Here X and Y are vectors such that X = (1001) and Y = (1010). Part b) Find a basis for W. The Attempt at a Solution So I know to satisfy being a...

Homework Statement Let M be a subspace of the vector space \mathbb{R}_2[t] generated by p_1(T)=t^2+t+1 and p_2(T)=1-t^2, and N be a subspace generated by q_1(T)=t^2+2t+3 and q_2(T)=t^2-t+1. Show the dimension of the following subspaces: M+N, M \cap N, and give a basis for each...

Hi, I have just begin with Linear Algebra. I came across cosets and I don't understand what is the difference between cosets and subspaces? thanx in advance.

Homework Statement What are the intersections of the following pairs of subspaces? (a) The x-y plane and the y-z plane in R'. (b) The line through (1, 1, 1) and the plane through (1,0, 0) and (0, 1, 1). (c) The zero vector and the whole space R'. (d) The plane S perpendicular to (1, 1...

Hello :) I've been doing a lot of work on subspaces but have come across this question and need a bit of help! Homework Statement W = {(x, y) \in R^{2} | x^{2} + y^{2} = 0} Homework Equations 1. 0 ∈ W 2. ∀ u,v ∈ W; u+v ∈ W 3. ∀ c ∈ R and u ∈ W; cu ∈ W The Attempt at a...

Homework Statement a) If U and W are subspaces of R^3, show that it is possible to find a basis B for R^3 such that one subset of B is a basis for U and another subset of B (possibly overlapping) is a basis for W. b) If U and W are subspaces of a finite-dimensional vector space V, show...

Homework Statement Let U1; U2 be subspaces of the vector space V . Prove that their intersection U1 \ U2 is also a subspace of V Homework Equations I see how any equations could be used here The Attempt at a Solution Well intuitively this seems obvious from the get go. If U1 and...

Homework Statement [PLAIN]http://i26.lulzimg.com/274748.jpg Homework Equations ?? The Attempt at a Solution i don't even know how to start. lol.

Let W1 and W2 be subspaces of a vector space V. Prove that W1\bigcupW2 is a subspace of V if and only if W1\subseteqW2 or W2\subseteqW1Well so far, I have proven half of the statement (starting with the latter conditions). Right now I'm struggling to show that the final conditions follow from...

Hi there. I started learning about subspaces in linear algebra and I came across a question which I'm unsure how to solve. I understand that there are 'rules' which need to be passed in order for something to be a subspace, but I have no idea how to start with this problem: Consider the set...

Prove that a vector space cannot be the union of two proper subspaces. Let V be a vector space over a field F where U and W are proper subspaces. I am not sure where to start with this proof.

If we want to caculate the projection of a single vector, v=(1,2) (which is an element of an R2 vector space called V) onto the subspace of V (which we call W), do we use projection of v onto W = <v,w1>w1 + <v,w2>w2 + ... <v,wn>wn However, if the individual values of v are not known (that...

Hi, I'm new to linear algebra. I'm pretty good at doing exercises with matrices and stuff but even though I've been looking in different books, looking all over the internet I can't get into vector spaces and subspaces. It seems like the books have some very elementary and simple examples and...

Homework Statement Let A be an m x n matrix such that the homogeneous system Ax=0 has only the trivial solution. a. Does it follow that every system Ax=b is consistent? b. Does it follow that every consistent system Ax=b has a unique solution? The Attempt at a Solution So if the homogeneous...

Homework Statement Suppose U and W are subspaces of V, and U \, \bigcup \, W is a subspace of V. Show that U \subseteq W. The Attempt at a Solution I have been working on this one for a bit and have not made any headway. I wish I could post anything, even a start to this one but I...

Homework Statement Prove that the intersection of any collection of subspaces of V is a subspace of V. Homework Equations To show that a set is a subspace of a vector space, I need to show that there exists an additive identity, and that the set is closed under addition and scalar...