Subspaces Definition and 334 Threads

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...

Homework Statement Learning about sums of subspaces and wanted to be sure that I am understanding this correctly. Say that I have two subspaces of R^2: U = {(x,y) in R^2 : y + 2x = 0} W = {(x,y) in R^2 : y - 3x = 0} and I wanted to geometrically (and algebraically) represent their...

Given a square matrix, if an eigenvalue is zero, is the matrix invertible? I am inclined to say it will not be invertible, since if one were to do singular value decomposition of a matrix, we would have a diagonal matrix as part of the decomposition, and this diagonal matrix would have 0 as an...

1. Let \mathbb{R}[x]_n^+ and } \mathbb{R}[x]_n^- denote the vector subspaces of even and odd polynomials in \mathbb{R}[x]_n Show \mathbb{R}[x]_n=\mathbb{R}[x]_n^+ \oplus\mathbb{R}[x]_n^- 3. For every p^+(x) \in \mathbb{R}[x]_n^+ \displaystyle p^+(x)=\sum_{m=0}^n a_m x^m=p^+(-x)...

Homework Statement Homework Equations In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system".[1] In more general...

If U, U ′ are subspaces of V , then the union U ∪ U ′ is almost never a subspace (unless one happens to be contained in the other). Prove that, if W is a subspace, and U ∪ U ′ ⊂ W , then U + U ′ ⊂ W . This seems fairly simple, but I am stuck on how to go about proving it.

Homework Statement http://img6.imageshack.us/img6/5017/69430037.th.png Uploaded with ImageShack.us The Attempt at a Solution a) http://www4d.wolframalpha.com/Calculate/MSP/MSP10519f5ffahf530d2ei00005867aii1c038ibgc?MSPStoreType=image/gif&s=29&w=178&h=56...

Homework Statement Let A be a fixed 2x2 matrix. Assuming that the set: W={X:AX=2X} has infinitly many solutions, determine whether it is a subspace of M(2,1) Homework Equations To determine whether a set is a subspace i need to prove that there is a zero vector, that it is closed under...

1) How to show that if W is a subspace of a finite-dimensional vector space V, then W is finite-dimensional and dim W<= dimV. 2) How to show that if a subspace of a finite-dimensional vector space V and dim W = dimV, then W = V. 3) How to prove that the subspace of R^3 are{0}, R^3 itself...

Homework Statement For each set below, which of the following sets are real linear subspaces (where addition and scalar multiplication are defined in the usual way for these sort of objects)? Justify your answers with an argument in which real linear space it is included, why it is closed...

Homework Statement Which of the following are a vector spaces? (a)R = {(a,b,c) 〖 ∈R〗^3 │ a+b=0 and 2a-b-c=0} (b)S = {(a,b)∈ R^2│ a^2=b^2 }..., can I say a =b which will make things simpler? (c)T = {(f∈P_2 (R)│ f(1)=f(0)+1} Homework Equations The Attempt at a Solution...

Homework Statement I'm given two subspaces L and K of P2 (R) are given by L = { f(x) : 19f(0)+f ' (0) = 0 } K = { f(x) : f(1) = 0 }. Obtain a non-trivial quadratic n = ax2 + b x +c such that n is element of the intersetion of L and K. Homework Equations The...

Homework Statement If V and W are 2-dimensional subspaces of \mathbb{R}^{4}, what are the possible dimensions of the subspace V \cap W? (A). 1 only (B) 2 only (C) 0 and 1 only (D) 0, 1, 2 only (E) 0,1,2,3, and 4 Homework Equations dim(V + W) = dim V + dim W - dim(V \cap W) dim (V + W) \leq...

Homework Statement This is just a conceptual question Whenever you are asked for a basis for the subspace spanner by some set of vectors, is that the same as asking the basis that forms the column space of that matrix? Are the dimension for that subspace the same as the column space...

hi.. how can we say a compact space automatically a locally compact? how subspace Q of rational numbers is not locally compact? am not able to understand these.. can anyone help me?

Homework Statement Give an example of a separable Hausdorff space (X,T) with a subspace (A,T_A) that is not separable. The Attempt at a Solution well since a separable space is one that is either finite or has a one-to-one correspondence with the natural numbers, the separable...

Homework Statement 1. Consider three linearly independent vectors v1, v2, v3 in Rn. Are the vectors v1, v1+v2, v1+v2+v3 linearly independent as well? 2. Consider a subspace V of Rn. Is the orthogonal complement of V a subspace of Rn as well? 3. Consider the line L spanned by [1 2...

Homework Statement If R3 is a vector space and V = (x,x,0) is a subspace, find unique subspaces W1 and W2 such that R3 = V ⨁ W1 = V ⨁ W2 Homework Equations The Attempt at a Solution Assuming R3 = (x,y,z) - please correct me if I'm wrong somehow - then I could pick a W1 like...

Homework Statement Determine in each case below whether U is a subspace V. If it is, verify all conditions for U to be a subspace, and if not, state a condition that fails and give a counter-example showing that the condition fails. There are parts (a) through (f) to this question. I am...

Hi, everyone: I have not found a clear explanation for this: Are any two contractible subspaces A,B<X homotopy-equivalent to each other? Clearly, homotopy equivalence is an equivalence relation. BUT: what if A<X is H-E (Homotopy-Equiv.) to a point p in X, while...

Homework Statement For a metric space (M,d) and two compact subspaces A and B define the distance d(A,B) between these sets as inf{d(x,y): x in A and y in B}. Prove that there exists an x in A and a y in B such that d(x,y)=d(A,B). Homework Equations The Attempt at a Solution I...

Homework Statement Hi, i am trying to do the question on the image, Can some one help me out with the steps. [PLAIN]http://img121.imageshack.us/img121/6818/algebra0.jpg Solution in the image is right but my answer is so off from the current one. Homework Equations The...

Heres the question: Let {u,v,w} be a linearly independent set of vectors of R^4. Let E = span{u,2v} and F=span{w,v}. Find EnF and E + F. i really have no idea other than i guess if 1/2u=w and v=v, then the EnF can be defined by that, but I'm not sure if that is right! :(

Homework Statement Which of these subsets of P2 are subspaces of P2? Find a basis for those that are subspaces. (Only one part) {p(t): p(0) = 2} Homework Equations The Attempt at a Solution So, I know the answer is that it's not a subspace via back of the book, but I don't...

Homework Statement [PLAIN]http://img571.imageshack.us/img571/1821/subspaces.png Homework Equations The Attempt at a Solution Is my solution correct?: For a,b\in \mathbb{C} let A=\begin{bmatrix} a \\ a \\ 0 \end{bmatrix}\in U and B=\begin{bmatrix} 0 \\ b \\ b...

Homework Statement Let F be the vector space (over R) of all functions f : R−R. Determine whether or not the following subsets of F are subspaces of F: Homework Equations 1. S1 = {f e F|f(−3) = 0 and f(10) = 0}; 2. S2 = {f e F|f(−3) = 0 or f(10) = 0}. The Attempt at a Solution I...

Homework Statement How many two dimensional subspaces does (F_3)^4 have? The attempt at a solution I chose an arbitrary basis so B = (v1,v2,v3,v4) for (F_3)^4 and then basically did 4C2 = 6 so it has 6 subspaces with dimension 2. However, thinking over this problem I've realized that I'm...

suppose v1 and v2 are two linear subspaces of a linear subspace v is there any measure of the distance between the two subspaces? in two dimensional complex space, i think the distance between x and y axes is the maximum possible value. Intuitively, if two subspaces are orthogonal to each...

Homework Statement Is U={f E F(\left|a,b\right|) f(a)=f(b)} a subspace of F(\left| a,b \right|) where F(\left| a,b \right|) is the vector space of real valued functions defined on the interval [a,b]?Homework Equations I know in order for something to be a subspace there are three conditions...

If U and V are vector subspaces of W and if U union V is also a subspace of W, how would i show that either U or V is contained in the other?

Hello! Just a quick question. Is the following okay?: The span of a set of vectors corresponds to a subspace in Rn. But the span of a set of vectors can also be ALL of Rn, does that mean all of Rn can be considered a subspace? Or does it mean the first definiton is not entirely correct, and...

Suppose V is a vector space over R and U_1, U_2, W are subspaces of V. Prove or contradict: 1. W \cap (U_1+U_2) = (W \cap U_1)+(W \cap U_2) 2. If\ U_1 \oplus W = U_2 \oplus W\ then\ U_1=U_2 I'm not sure how to approach this problem, and will appreciate any guidance. Thanks! (For the...

I have a Hilbert space H; given a closed subspace U of H let PU denote the orthogonal projection onto U. I also have a lattice L of closed subspaces of H, such that for all U and U' in L, PU and PU' commute. The problem is to find an orthonormal basis B of H, such that for every element b of B...

This is an exercise in a linear algebra textbook that I initially thought was going to be easy, but it took me a while to make the proof convincing. Prove: Any subspace of a finite-dimensional vector space is finite-dimensional. Here's my attempt. I am not sure about some details and I'm...

Find a basis for the given subspaces of R3 and R4. a) All vectors of the form (a, b, c) where a =0. My attempt: I know that I need to find vectors that are linearly independent and satisfy the given restrictions, so... (0, 1, 1) and (0, 0, 1) The vectors aren't scalar multiples of...

Hi, I'm trying to solve this exercise. Homework Statement Given the following subsets of \mathbb(R)^3 find the subspaces generated by them: \{(1,0,-2),(-1,0,2),(3,0,-1),(-1,0,3)\} The Attempt at a Solution I've tried to solve the linear dependence, so I've made the system: \begin{Bmatrix}...

Homework Statement Find the T-Cyclic subspace generated by Z. V = P1(\Re) T(f) = f' +2f and Z = 2xHomework Equations The Attempt at a Solution so T(1,0) = 2 and T(0,1) = 1 + 2x so [T]_{}\beta = ( 2 1 0 2 ) So T-cyclic subspace generated by 2x = { 2x, 2 + 4x } ?

Homework Statement Show that W is a T-invariant subspace of T for: W = E_{\lambda}Homework EquationsThe Attempt at a Solution Ok, so I know that I need to show that T maps every element in E_{\lambda} to . E_{\lambda} = N(T-\lambdaI) so T must map every eigenvector related to \lambda to...

If N>2 and A\in\textrm{SO}(N) are arbitrary, does there exist subspaces V_1,V_2\subset\mathbb{R}^N such that V_1+ V_2 = \mathbb{R}^N,\quad\quad \textrm{dim}(V_1)=2,\quad \textrm{dim}(V_2)=N-2 and such that the restriction of A to V_1 belongs to \textrm{SO}(2), and that the restriction of A...

Let S= { v1, v2, v3...vn} be a set of vectors in a vector space V and let W be a subspace of V containing S show W contains span S. Span is the smallest subspace (w) of vector space V that contains vectors in S if a and b are two vectors in subspace C, then they are linear combinations...

Homework Statement Prove that all bases for subspace V of R\hat{}N contain the same number of elements. Homework Equations The Attempt at a Solution I have absolutely no idea where to start this proof. Do I need to do something with finding an equation of the subspace, or not?

Homework Statement For each linear operator T on the vector space V, find an ordered basis for the T-Cyclic subspace generated by the vector z. a) V = R4, T(a+b,b-c,a+c,a+d) and z= e1 Homework Equations Theorem: Let T be a linear operator on a finite dimensional vector space V, and let...

Homework Statement Let T be a linear operator on a vector space V and let W be a T-Invariant subspace of V. Prove that W is g(T)-invariant for any polynomial g(t). Homework Equations Cayley-Hamilton Theorem? The Attempt at a Solution Im not sure how to begin. Ok so g(t) is the...

Ok so I've been working on this problem and I'm really having some struggles grasping it. Here it is: Let W be some subspace of Rn, let WW consist of those vectors in Rn that are orthognoal to all vectors in W. 1) Show that WW is a subspace of Rn? So for this part I'm thinking that...