Subspace Definition and 560 Threads

Let S be a subspace of R3 spanned by u2=\left[ \begin{array} {c} \frac{2}{3} \\ \frac{2}{3} \\ \frac{1}{3} \end{array} \right] and u3=\left[ \begin{array} {c} \frac{1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} \\ 0 \end{array} \right]. Let x=\left[ \begin{array} {c} 1 \\ 2 \\ 2 \end{array} \right]...

Homework Statement Prove that V is a subspace of R4 Actual problem is attached Homework Equations - S contains a zero element - for any x in V and y in V, x + y is in V - for any x in V and scalar k, kx is in V The Attempt at a Solution Its obvious that V is a subset of R4...

Homework Statement Show that W ={(x,y,z) : x +2y +3z =0} is a subspace of R3. By finding a subset S of W such that span(S) = W.Homework Equations Ab = X?The Attempt at a Solution I don't have an attempt because I'm completely lost where to start . Can someone point me in the right way please.

1. Let U be a subspace of Rn and let U⊥ = {w ∈ Rn : w is orthogonal to U} . Prove that (i) U⊥ is a subspace of Rn, (ii) dimU + dimU⊥ = n. Attempt. i) U. ( U⊥)T=0 If U⊥ does not passes the origin , the above equation cannot be satisfied. Therefore U⊥ passes the origin. U.( U⊥+...

H = ([a,b;c,d] : a+d =0} Dim(M2x2)= 4, so a basis would have 4 components? I got this far and am stuck. [a, b ; c , -a] = a[1,0; 0,-1] + b[0, 1;0,0] +c[0,0; 1,0]

Homework Statement P={(x,y,z)|x+2y+z=6}, a plane in R3. P is not a subspace of R3. Why? Homework Equations See below. The Attempt at a Solution I am really quite confused here. My text says: "A subset W of a vector space V is called a subspace of V if W is itself a vector space...

The set of all functions f such that the integral of f(x) with respect to x over the interval [a,b] is 1. equal to zero 2. not equal to zero 3. equal to one 4. greater than equal to one etc. How can we determine this types of set is a subspace of not. for the case of the set of...

So an example was the matrix: A = \left(\begin{array}{cccc} a&a+b\\ b&0\\ \end{array} \right) is a subspace of M2x2. and is the linear combination a*\left(\begin{array}{cccc} 1&1\\ 0&0 \end{array} \right) + b*\left(\begin{array}{cccc} 0&1\\ 1&0 \end{array} \right) Meaning it has...

Homework Statement Suppose V is a vector space with operations + and * (under the usual operations) and W = {w1, w2, ... , wn} is a subset of V with n vectors. Show Span{W} is a subspace of V. The attempt at a solution I know that to show a set is a subspace, we need to show...

Homework Statement Determine whether the following sets form subspaces of R2: { (X1, X2) | |X1| = |X2| } { (X1, X2) | (X1)^2 = (X2)^2 } Homework Equations The Attempt at a Solution I'm clueless. I've been trying to figure it out for a good thirty minutes on both of them, but I'm...

Homework Statement Let 'S' be the collection of vectors [x;y;z] in R3 that satisfy the given property. Either prove that 'S' forms a subspace of R3 or give a counterexample to show that it does not. |x-y|=|y-z| Homework Equations The Attempt at a Solution First I tested the 0...

Homework Statement Simple enough: Is P2 a subspace of P3? Homework Equations The Attempt at a Solution I think it is. All P2's can be written in the form 0x^3 + ax^2 + bx + c. Then, it's easy to see that it's closed under scalar addition and multiplication. Our professor...

C[-pi, pi] Span(1, cos (2x), cos2 (x)) Doing the Wronskian here is pain so what other method would be more appropriate?

Show that C^n[a,b] is a subspace of C[a,b] where C^n is the nth derivative. I know the set is non empty since f(x)=x exist; however, I don't know how to start either the multiplication or addition property of subspaces to confirm that C^n is a subspace. Thanks ahead of time for any help...

1. True or False: If U is a subspace of V, and V is a subspace of W, U is a subspace of W. If true give proof of answer, if false, give an example disproving the statement. 2. My thoughts: If U is a subspace of V, then the zero vector is in V. As well as x+v is in V and ax is in V (by...

The set of all functions f in C[-1,1], f(-1)=0 and f(1)=0. Nonempty since f(-1) = x^(2n) - 1 and f(1) = x^n - 1 ϵ C[-1,1] where n ϵ ℤ, n ≥ 0 α·x^(2n) - α = α (x^(2n) - 1) = α·0 = 0 and α·x^n - α = α (x^n - 1) = α·0 = 0 x^(2n) - 1 + x^n - 1 = (x^(2n) - 1) + (x^n - 1) = 0 + 0 = 0 Based...

Homework Statement Prove or disprove this with counter example: Let U,V be subspaces of R^n and let B = {v1, v2,...,vr} be a basis of U. If B is a subset of V, then U is a subset of V. Homework Equations U and V are subspaces so 1. zero vector is contained in them 2. u1 + u2 is...

Hi I am presented with the following problem Homework Statement Let P_{4}(\mathbb{R}) = \{a_0 + a_1 \cdot x + a_2 \cdot x^2 + a_3 \cdot x^3|a_0,a_1,a_2,a_3 \in \mathbb{R} \} Then let S be a subset of P_4(\mathbb{R}) where S = \{a_0 + a_1 \cdot x + a_2 \cdot x^2 + a_3 \cdot...

Question: In R3, show that (1,-1,0) and (0,1,-1) are a basis for the subspace V={(x,y,z) \in R3: x+y+z=0} Attempt: By def of a basis, the vectors (1) must be linearly independent and (2) must span V. 1. For LI, show that if a(1,-1,0) + b(0,1,-1) = (0,0,0), then a=b=0...

Homework Statement Here's a statement, and I am supposed to show that it holds. If x,y, and z are vectors such that x+y+z=0, then x and y span the same subspace as y and z. Homework Equations N/A The Attempt at a Solution If x+y+z=0 it means that the set {x,y,z} of vectors...

Let V be a 9 dimensional vector space and let U and W be five dimensional subspaces of V with the bases Bu and Bw respectively, (a) show that if Bu intersect Bw is empty then Bu union Bw is linearly dependen (b)use part (a) to prove U intersect W is not equal to the 0 vector now i have...

Homework Statement V = the set of all symetrical nXn matrices, A=(ajk) such that ajk=akj for all j,k=1,...,n Determine the base and dimensions for V The Attempt at a Solution I set my matrix up as [a11 a12] [a21 a22] So a21 and a12 are equal to each other? I assume the...

Homework Statement If U and V are subsets of R^n, then the set U+V is defined by U+V={x:x=u+v,u in U, and v in V} prove that U and V are subspaces of R^n then the set U+V is a subspace of R^n. I am just having trouble proving U+V is a subspace. Homework Equations To be a...

My book made the following claim... but I don't understand why it's true: If v_1, v_2, v_3, v_4 is a basis for the vector space \mathbb{R}^4, and if W is a subspace, then there exists a W which has a basis which is not some subset of the v's. The book provided a proof by counterexample...

Homework Statement Suppose that V is a vector space, W and X are subspaces with X contained in W. Show that there is a subspace U of V which is maximal subject to the property that U intersect W equals X.Homework Equations N/AThe Attempt at a Solution I know this uses Zorn's Lemma but I can't...

we sway into scifiction which i want to be realistic: How is it inside Subspace or Hyperspace or a cosmic string (from the perspective of different theories)? Can we even enter cosmic strings (if we say we had the technology)? Has the subspace (enbedded in our 4 dimensions) or the...

say we are given a subspace like this: Being W the subspace of R generated by (1,-2,3,-1), (1,1,-2,3) determine a basis and the dimension of the subspace. Won't the vectors given work as a basis, as long as they are linearly independent? If so, all we have to do is check for dependance, and if...

so, if I want to calculate the subspace spanned by A in: A = {(1,0,1) , (0,1,0)} in R^{3} c_{1}(1,0,1)+c_{2}(0,1,0) = (x,y,z) i can make a system: c_{1} = x c_{2} = y c_{1} = z from which I can conclude that x = z, and so, the subspace spanned will be the plane given by x =...

Homework Statement Assume that e_1 ,..., e_n is a basis for the vector space V. Let W be the linear subspace determined (formed?) by the vectors e_{1}-e_{2}, e_{2}-e_{3}, ..., e_{n-1}-e_{n}, e_{n}-e_{1}. Determine the dimension of W, and a basis for W. Homework Equations The...

There's not really a problem statement here. I just want to know : If I have a vector starting on the origin (like a position vector), then it will always correspond to a vectorial subspace, right? For example: (b, 2a + b ) : a, b \in R is a vectorial subspace but is (b, 2a +...

in a space V^n, prove that the set of all vectors {v1,v2,..}, orthogonal to any v≠0, form a subspace V^(n-1). i know that a subspace of V^n must be at least one dimension less and the set of vector v1,v2,... build a orthogonal basis, but how can one show with this preconditions that the...

Suppose I have 3 vectors e1, e2, e3 that spans the subspace E, another 3 vectors d1, d2, d3 that spans the subspace D. If I also know that e1’d1 = 0, e2’d2 = 0, e3’d3 = 0, are there any conclusions I can make in terms of E and D? like row(E) = null(D)?

1. Let W be the linear subspace spanned by the polynomials 1 and x. Find an orthogonal projection of the polynomial p(x) = 1+x^2 to W. Find a basis in the space W(perp) My problem is that I don't know how to represent W as a matrix so that I could apply the orthogonal projection formula...

Homework Statement Determine whether or not the set of vectors: \left\{\bar{x}=t \left( \begin{array}{cc}1\\2\\1\end{array} \right) +s\left( \begin{array}{cc}1\\1\\1\end{array} \right) +\left( \begin{array}{cc}1\\0\\2\end{array} \right),-\infty < t,s < \infty \right\} is a...

Homework Statement We are given a subspace of R^3 that is produced by the elements: (2,6,2) abd (6,2,2). We are asked to find (if any) a homogeneous linear system that has this subspace as solution set. Homework Equations The Attempt at a Solution 1)The subspace is 2...

I just read Ascoli's theorem: A subspace F of C(X,R^n) has compact closure if and only if F is equicontinuous and pointwise bounded. Then it says, As a corollary: If the collection {fn} of functions in C(X,R^k) is pointwise bounded and equicontinuous, then the sequence (fn) has a uniformly...

Homework Statement Find a basis of the subspace of R4 that consists of all vectors perpendicular to both (1 0 5 2) and (0 1 5 5) ^ those are vectors. Homework Equations The Attempt at a Solution I understand that a basis needs to be linearly independent and...

Let S be a subspace of L^{2}(\left[0,1\right]) and suppose \left|f(x)\right|\leq K \left\| f \right\| for all f in S. Show that the dimension of S is at most K^{2} --------- The prof hinted us to use Bessel's inequality. Namely, let \left\{ u_1,\dots, u_m \right\} be a set of...

Homework Statement Denote by W the set of all vectors that are of the form x = (a, b 2a, 3b,-a), in which a and b are arbitrary real numbers. Show that W is a linear subspace of R^5. Also find a set of spanning vectors for W. What kind of geometric object is W? Homework Equations...

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...