Subspace Definition and 560 Threads

suppose that {ei} 1<=i<=n is a basis for v and w is a subspace of v of dimension m<n ... . so Can we always find a basis for w which includes m elements of {ei} 1<=i<=n ? does always w include m elements of {ei} 1<=i<=n ?

Given a subset W of a vector space V = F^n (for some field F), prove that W is the subspace of solutions of the matrix equation AX = 0 for some A.

Please help? I have done all the parts of question 1 but i really can't solve (iv) part. i know that the answer should be [-8;8;-1;0;1]. if someone can pleaseeeeeeeeeeeee help me? thank you very much Maria! http://img3.imageshack.us/img3/5706/72162896rd8.jpg

Subspace of a Function?!? Homework Statement {f \in C([0, 1]): f(1/2) = 0} Is this subset of C([0,1]) a subspace? Homework Equations C[0,1] be the set of all functions that are continuous on [0, 1]. (f + g)(x) = f(x) + g(x) (af)(x) = a*f(x) The Attempt at a Solution...

in linear algebra, if i am told to find a basis for the following W={(x,y,z,t)|x+y=0, x+t=0} what i did was 1 1 1 0 0 0 0 1 after performing elementary actions on rows, i came to 1 0 0 1 0 0 0 0 from here i can see that they are linearly independant and they cleary span...

Homework Statement Find a basis for the subspace S = {(a+2b,b-a+b,a+3b) | a,b \in R } \subseteq R^4 What is the dimension of S? Homework Equations The Attempt at a Solution a(1,0,-1,1) + b(2,1,1,3) , a,b \in R span { (1,0,-1,1) , (2,1,1,3) } So I put (1,0,-1,1) as V1...

Homework Statement Show that the following set of vectors are subspaces of R^m The set of all vectors (x,y,z) such that x+y+z=0 of R^3 . Then find a set that spans this subspace. Homework Equations The Attempt at a Solution I managed to proof that the set of vectors is a...

If a set of vectors does not contain the zero vector is it still a subspace?

i know that in order to prove that one group of vectors are a part of another i need to stack them up i did row reduction and i don't know how to extract a vector for the group http://img384.imageshack.us/img384/2546/55339538nk4.gif this came from this question part 2...

Homework Statement Determine whether the subset W of the vector space V is a subspace of V. Let V = L(Q4) (the set of linear transformations from rational numbers with 4 coordinates to rational numbers with 4 coordinates). Let W = { T in V = L(Q4) | { (1,0,1,0) , (0,1,0,-1) } is contained in...

Homework Statement In this case find the basis of subspace U 1 2 3 4 5 6 7 8 -6 -8 -10 12 Homework Equations elementary row operations The Attempt at a Solution alright, so i know i have to reduce the matrix and i have done so 1 2 3 4 0 1 2 3 0 0 0 1 now the answer...

Homework Statement Determine whether U is a subspace of R3 U= [0 s t|s and t in R] Homework Equations My textbook, which is vague in its explinations, says the following "a set of U vectors is called a subspace of Rn if it satisfies the following properties 1. The zero vector 0 is in...

Homework Statement Find a basis for the subspace of R3 spanned by S={(42,54,72),(14,18,24),(7,9,8)}. I am not sure what steps to take to solve this. Any help would be great.

Homework Statement Claim: Matrices of trace zero for a subspace of M_n (F) of dimension n^2 -1 where M_n (F) is the set of all nxn matrices over some field F. Homework Equations Tr(M_n) = sum of diagonal elements The Attempt at a Solution I view the trace Tr as a linear...

My question is; Let S = {A € Mn,n | A = AT } the set of all symmetric n × n matrices Show that S is a subspace of the vector space Mn,n I do not know how to start to this if you can give me a clue for starting, I appreciate.

Homework Statement Let B be a fixed n x n matrix, and let X_B = { A e M_n so that AB = BA }. In other words, X_B is the set of all matrices which commute with B. (a) Prove that X_B is a subspace of M_n. (b) Let B = [ 1 0 2 -1 ] Find a basis for X_B and write its dimension. (c)...

Homework Statement S is a subset of vector space V. If V = 2x2 matrix and S ={A | A is invertible} a) is S closed under addition? b) is S closed under scalar multiplication? Homework Equations The Attempt at a Solution For non singular 2x2 matrices, S is not closed under...

Homework Statement S is a subset of vector space V, If V is an 2x2 matrix and S={A|A is singular}, a)is S closed under addition? b) is S closed under scalar multiplication? Homework Equations S is a subspace of V if it is closed under addition and scalar multiplication...

Homework Statement is B in subspace R^2 B=[x] :x^2+y^2<=1 [y] Homework Equations 1.0∈B 2.if u,v∈B u+v∈B 3.if u∈A a∈B then au∈B The Attempt at a Solution The <= is confusing me. I am not sure if i am suppose to treat it like an equal sign, or is it automatically not in the...

Alright so I am trying to find the projection matrix for the subspace spanned by the vectors [1] and [2] [-1] [0] [1] [1] I actually have the solution to the problem, it is ... P = [ 5 1 2 ] (1/6) [1 5 -2]...

missed last week due to illness so have no clue what this homework is going on about, the question is for each of the following state whether or not it is a subspace of R3, Justify your answer by giving a proof or a counter example in each case, i know I'm ment to attempt the question before i...

I got a small test tomorrow and i have been working throu exercises but i can't seem to solve this question: Let V be a vector space over a field F, and let S\subset S' be subsets of V. a) Show that span(S) is a subspace of V. b) Show that span(S) is a subset of Span(S'). c) Take...

Homework Statement If \vec{x} is an eigenvector of a Hermitian matrix H, let V be the set of vectors orthogonal to \vec{x} . Show that V is a subspace, and that it is an invariant subspace of H. The Attempt at a Solution The Hermitian H must act on some linear space, call it K and of...

Use the subspace Theorem to decide if the following are subspaces of P2, the vector space of all polynomials of degree at most 2. a) R1 = {ao + a1x +a2x^2 | ao = 0} b) R1 = {ao + a1x +a2x^2 | a1 = 1} c) R1 = { p E P2 | p has exactly degree 2} (for part c 'E' is 'element of')...

Consider the prime field F_p p a prime. How can I count the number of subfields there are? Is this a known result of algebra?

Homework Statement Find a basis for the subspace S of R^4 consisting of all vectors of the form (a+b, a-b+2c, b, c)^T, where a,b,c are real numbers. What is the dimension of S? Homework Equations vectors v1,...,vn from a basis for a vector space iff i) v1,...,vn are linearly...

Consider the prime field F_p p a prime. How can I count the number of subfields are there are? Is this a known result of algebra?

Homework Statement The Attempt at a Solution I don't really know how to do this, so I hope someone can give some hints or briefly tell me what I should do.

I was revising linear algebra and came across the topic of 'constructing complementary subspace given a subspace' - and since the proof (that used Zorn's lemma) of its (complementary subspace's) existence was not constructive, the author defined an equivalence relation in constructing a...

Let \vec{u},\vec{v},\vec{w} be fixed vectors in Rn. Define S to be the set of all vectors in Rn which are linear combinations of the form k_1 \vec{u}+k_2 \vec{v}+3 \vec{w}, where k_1,k_2 \in R. Is S a subspace of Rn? Im a little stuck with this one. I've tried defining two vectors...

Let A,B be n x n matrices and e1 be the first standard basis vector in Rn. For each of the following subsets of Rn, determine whether it is a subspace of Rn, giving reasons. \begin{array}{l} \left\{ {x \in R^n |Ax = 2x} \right\} \\ \left\{ {x \in R^n |Ax = 2x + e_1 } \right\} \\...

Let \vec{a} \ne 0 be a fixed vector in R3. (a) Prove that the set of all vectors \vec{x} \in R^3 satisfying \vec{a}.\vec{x}=0 is a subspace of R^3. Describe this set geometrically. (b) Is the set of all vectors \vec{x} \in R^3 satisfying \vec{a} \times \vec{x}=0 a subspace of R3?For part (a)...

Not sure how to prove the following: If U is a subspace of a vector space V, and if u and v are elements of V, but one or both not in U, can u + v be in U? Any help would be appreciated.

http://www.mjyoung.net/misc/quantum.htm read this and tell me if it works or if its a crock, it sounds good, but all I've had is high school Physics

[SOLVED] Caracterizing a subspace of L^2 Homework Statement Call M the subspace of L²([0,1]) consisting of all functions of vanishing mean. I.e., u\in M \Rightarrow \int_0^1u(s)ds=0. I am trying to find the dimension of the orthogonal of M, M^{\perp}=\{x\in...

[SOLVED] Show subspace of H^1[0,1] is closed Homework Statement I have an assignment that deals with some Sobolev spaces but I have never worked with them before. Only the definitions are given. Consider the Sobolev space W^{1,2}([0,1])=H^1([0,1])=\{u\in C([0,1]): \mbox{ there exists }...

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!