Subspace Definition and 574 Threads

Homework Statement Suppose T is a linear operator on a finite dimensional vector space V, such that every subspace of V with dimension dim V-1 is invariant under T. Prove that T is a scalar multiple of the identity operator. The Attempt at a Solution I'm thinking of starting by letting U...

Homework Statement Check if this sets of vectors generate same subspace for \mathbb{R}^4. { (1,2,0,-1), (-2,0,1,1) } and { (-1,2,1,0),(-3,-2,1,2) , (-1,6,2,-1) } Homework Equations The Attempt at a Solution Here is the one matrix. \begin{bmatrix} 1 & 2 & 0 & -1 \\ -2 & 0 & 1 & 1...

Let V be the vector space of all functions from R to R, equipped with the usual operations of function addition and scalar multiplication. Let E be the subset of even functions, so E = {f \epsilon V |f(x) = f(−x), \forallx \epsilon R} , and let O be the subset of odd functions, so that O = {f...

I have a question about Lie subalgebra. They say "a Lie subalgebra is a much more CONSTRAINED structure than a subspace". Well, it seems subtle, and I find this very tricky to follow. Can anyone explain this with concrete examples? If my question is not clear, please tell me so, I will...

Homework Statement For each s \inR determine whether the vector y is in the subspace of R^4 spanned by the columns of A where y= 6 7 1 s and A = 1 3 2 -1 -1 1 3 8 1 4 9 3 Homework Equations The Attempt at a Solution Can i do this by making an...

use rowspace/colspace to determine a basis for the subspace of R^n spanned by the given set of vectors: {(1,-1,2),(5,-4,1),(7,-5,-4)} *note: the actual instructions are to use the ideas in the section to determine the basis, but the only two things learned in the section are rowspace and...

The question I am looking at asks, Let a be the vector (-1,2) in R^2. Is the set S = { x is in R^2 | x dot a = 0} a subspace? --> x and a are vectors... Can anyone explain how to show this? I was thinking that since the zero vector is in R^2, this must also be a subspace...

Homework Statement Let n \geq2. Which of the conditions defining a subspace are satisfied for the following subsets of the vector space Mnxn(R) of real (nxn)-matrices? (Proofs or counterexamples required. There are three subsets, i will start with the one where The subset V is that of...

Homework Statement Let (F2) ={0,1} denote the field with 2 elements. i) Let V be a vector space over (F2) . Show that every non empty subset W of V which is closed under addition is a subspace of V. ii) Write down all subsets of the vector space (F2)^2 over (F2) and underline those...

Homework Statement Determine whether the following is a subspace of P_{4}_ (a) The set of polynomials in P_{4} of even degree. Homework Equations P_{4} = ax^{3}+bx^{2}+cx+d The Attempt at a Solution (p+q(x)) = p(x) + q(x) (\alpha p)(x) = p(\alpha x) If p and q are both of...

Hi all, I have a problem related to quantum mechanical description of vibrational motion in molecules. I would like, for efficiency, to integrate over symmetries (global rotations) of the molecule. I would like to prove (or disprove) that all points in R^N can be reached by rotations from a...

Homework Statement Are the vector subspaces U={(x,y,0,0) | x+2y=0} and W= {(0,0,z,t | z+t=0} from R^4 stand for U+W = R^4Homework Equations The Attempt at a Solution Can somebody explain, how will solve this task. I have no idea, how they do in my book. Thanks.

Hi! I just want to ask you, what is the principle of finding section between two vector subspaces. Let's say: U={(a,b,0) | a, b \in R} and W={(a,b,c) | a+b+c=0, a,b,c \in R} U, W are vector subspaces from R\stackrel{3}{}. P.S this is not homework question, just example, for my better...

What would be an example of two closed subspaces of a normed (or Banach) space whose sum A+B = {a+b: a in A, b in B} is not closed? I suppose we would have to look in infinite dimensional space to find our example, because this is hard to imagine in R^n!

I am curious as to why a subset of a vector space V must have the vector space V's zero vector be the subsets' zero vector in order to be a subspace. Its just not intuitive.

1. The set of all traceless (nxn)-matrices is a subspace sl(n) of (bold)K^(nxn). Find a basis for sl(n). What is the dimension of sl(n)? Not sure how to go about finding the basis. I know a basis is a list of vectors that is linearly independent and spans. and for the dimension of sl(n), is...

[SOLVED] Finding a basis for a subspace Homework Statement We have a subspace U in R^3 defined by: U = {(x_1 , x_2 ; x_3) | x_1 + 2*x_2 + x_3 = 0 }. Find a basis for U. The Attempt at a Solution We have the following homogeneous system: (1 2 1 | 0). From this I find the...

What would be an example of a not (topologically) closed subspace of a normed space?

V is a subspace of R^4 V={(x, -y, 2x+y, x-2y): x,y E R} 1) extend {(2,-1,5,0)} to a basis of V. 2) find subspace W of R^4 for which R^4= direct sum V(+)W. solution: 1)the dimension of V is 2.therefore i need to add one more vector to (2,-1,5,0). the 2nd vector is (1,0,2,1)...

Homework Statement In a space V^{n} , prove that the set of all vectors \left\{|V^{1}_{\bot}> |V^{2}_{\bot}> |V^{3}_{\bot}> ... \right\} orthogonal to any |V> \neq 0 , form a subspace V^{n-1} Homework Equations The Attempt at a Solution I tried to make a linear combination from...

Could Quantum Foam be used as a type of subspace, like the kind used in Star Trek? Mainly for communication...

Let u and v be (fixed) vectors in the vector space V. Show that the set W of all linear combinations au+bv of u and v is a subspace of V. I cannot prove the above proof properly. Can anyone help. -Thanks

Let A be an nxn matrix and let H= {B E Mnxn|AB=BA}. Determine if H is a subspace of Mnxn. This was a test question that I got incorrect. I didn't like the way my teacher proved this afterwards, they said it IS a subspace of Mnxn. Any help in explaining how it could be would be greatly...

Well it's not homewrok cause i don't need to hand this question in, this is why i decided to put it here. (that, and there isn't a topology forum per se, perhaps it's suited to point set topology so the set theory forum may suit it). Now to the question: Show that if Y is a subspace of X...

W={(x1,x2,x3):x^{2}_{1}+x^{2}_{2}+x^{2}_{3}=0} , V=R^3 Is W a subspace of the vector space? from what i understand for subspace to be a subspace it has to have two conditions: 1.must be closed under addition 2.must be closed under multiplication so... I pick a vector s=(s1,s2,s3) and a second...

Let A be a fixed 2x3 matrix. Prove that the set W={x \in R^{3} :Ax=[^{1}_{2}]} (2x1 matrix 1 on top 2 at the bottom) what does the information after the ":" mean? is it a condition? I don't understand this problem. Can anyone help me out?

Hi all, I have some questions about the concept of subspace of linear transformation and its dimension, when I try to prove following problems: Prove T is a finite dimensional subspace of L(V) and U is a finite dimensional subspace of V, then T(U) = {F(u) | F is in T, u is in U} is a...

when dealing with vector subspaces, an example of a question asks: if x=(a,a,b)T (T is really a superscript) is any vector in S, then... what does that T superscript notation mean? does that mean transpose? Thanks

Algebra- Vector space and subspace Homework Statement Here are some true or false statements given in my test. (a) R^2 is a subspace of R^3. (b) If {v1, v2, ..., vn} is a set of linearly dependent vectors, then it contains a zero vector. (c) If {v1, v2, ..., vn} is a spanning set, then...

Prove that the intersection of any collection of subspaces of V is a subspace of V. Ok, I know I need to show that: 1. For all u and v in the intersection, it must imply that u+v is in the intersection, and 2. For all u in the intersection and c in some field, cu must be in the...

Homework Statement Which of the following subsets of R^n*n are in fact subspaces of R^n*n 1) The symmetric matrices 2) The diagonal matrices 3) The nonsingular matrices 4) The singular matrices 5) The triangular matrices 6) The upper triangular matrices 7) All matrices that commute...

Is every subspace of a connected space connected?

I am drawing some strange mental blank with one question in my final exam review. Homework Statement V is the set of all polynomials that are of the form p(x) = cx^2 + bx+a U is a subset of V where all members satisfy the equation p(5) =0 Find a basis for U. I am not sure why I...

Anton, H. Elementary linear algebra (5e, page 156) says: If W is subset of a vector space V, then W is a subspace of V if and only if the following TWO conditions hold 1) If u and v are vectors in W, then u+v is in W 2) If k is any scalar and u is any vector in W, then ku is in W...

Is there any difference between a vector subspace and a linear manifold. Paul Halmos in Finite Dimensional Vector Spaces calls them the same thing. Hamburger and Grimshaw in Linear Trasforms in n Dimensional Vector Space does not use the word subspce at all. Planet Math says a Linear...

How would I prove this theorem: "The column space of an m x n matrix A is a subspace of R^m" by using this definition: A subspace of a vector space V is a subset H of V that has three properties: a) the zero vector of V is in H. b) H is closed under vector addition. c) H is closed under...

Homework Statement Let A = [2 -1 9] [1 0 5] [0 1 1] Find solution space W and prove that W is a subspace of R^3. Homework Equations The Attempt at a Solution rref= [1 0 5] [0 1 1] [0 0 0] So I know the row-echeleon form, which is what I suppose is the solution...

Find the dimension of the subspace spanned by the vectors u, v, w in each of the following cases: i) u = (1,-1,2)^T v = (0,-1,1)^T w = (3,-2, 5)^T ii) u = (0,1,1)^T v = (1,0,1)^T w = (1,1,0)^T Right, how do I go about this, do I have to find the subspace first then do the dimension. Can...

I am having some trouble with the following linear algebra problems, can someone please help me? 1) Explain what can be said about det A (determinant of A) if: A^2 + I = 0, A is n x n My attempt: A^2 = -I (det A)^2 = (-1)^n If n is be even, then det A = 1 or -1 But what happens when n...

Q: Is the subset a subspace of R3? If so, then prove it. If not, then give a reason why it is not. The vectors (b1, b2, b3) that satisfy b3- b2 + 3B1 = 0 ----------------------- My notation of a letter with a number to the right, (b1) represents b sub 1. Im having a problem on how...

Is the basis for the subspace W of a vectorspace V spanned by the basis of the vectorspace V? If so how?

Homework Statement choose x = (x1, x2, x3, x4) in R^4. It has 24 rearrangements like (x2, x1, x3, x4) and (x4, x3, x1, x2). Those 24 vectors including x itself span a subspace S. Find specific vectors x so that dimS is 0, 1, 3, 4 The Attempt at a Solution So, i thought of it this way: 24...

I want to confirm something: what is the smallest subspace of 3x3 matrices that contains all symmetric matrices and lower triangular matrices? - identity(*c)? because that is the only symmetric lower triangular i could think of... what is the largest subspce that is contained in both of...

Let V be a vector space over a field F, and M a subspace of V, where M is not {0}. I need to show there exists a basis for V such that none of its elements belong to M. Since M is a subspace of V, M must be a subset of V. If M = V, then there does not exist such a basis, so M must be a...

Let U be a proper subspace of R^4 and let it be given by the equations: 1) x1+x2+x3+x4=0 2) x1-x2+2x3+x4=0 how do i find a basis for this subspace? I got that (0,1,2,0) is one of the basis vectors since x2=2x3, therefore whatever we pick for x2, x3 will be twice that value. i also...

I have a question that states: Let a, b, and c be scalers such that abc is not equal to 0. Prove that the plane ax + by + cz = 0 is a subspace of R^3.

I am not sure about these 2 whether they are subspaces or not (i do know how to check whether it is a subspace or not) subsets of R^3, subspace or not? 1.all combinations of (1,1,0) and (2,0,1) 2.plane of vectors (b1,b2,b3) that satisfy b3-b2+3b1 = 0 thanks for help.

I think R2 is a subspace of R3 in the form(a,b,0)'.

Hi: was wondering if somebody can help me with this I came across in a paper. $e_k$ is a vector of $k$ $1's. M is a matrix of size n \times k. The author talks about projecting $M$ onto the null space of $e_k$. This is what confuses me. Which $x$ apart from the 0-vector solves e_kx=0...

I'm preparing for an end of year linear algebra test which briefly covers things about subsets of Matnxn and their relations to subspaces of Matnxn; I found the following question: Which of the following subsets of Matn×n are subspaces of Matn×n? (i) Symmetric matrices (i.e. matrices A...