Subspace Definition and 574 Threads

Homework Statement Which of the subsets of P2 given in exercises 1 through 5 are subspaces of P2? Find a basis for those that are subspaces. (P(t)|p(0) = 2) Homework Equations The Attempt at a Solution The solution manual says that this subset is not a subspace because it...

The subspace formed by matrix A and A' will be same or different if A' is obtained by applying an elementary row operation on A? Please prove it.

So I'm trying to get an idea of what an invariant subspace is and so please let me know if my understanding is correct. Given that you have some vector subspace being a collection of a particular number of vectors with the the space denoted as |\gamma>. If you have some other collection of...

The question is at the end of a chapter on spanning vector spaces. Homework Statement Let P denote an invertible n x n matrix. If \lambda is a number, show that E_{\lambda}(PAP^{-1}) = \left\{PX | X\;is\;in\;E_{\lambda}(A)\right\} for each n x n matrix A. [Here E_{\lambda}(A)} is...

Here's one I've been stewing over: - Let S be a nonempty set of F, and F a field. - Let F(S,F) be the set of all functions from S to the field F. - Let C(S,F) denote the set of all functions f \in F(S,F), such that f(s) = 0 for all but a finite number of elements in S (s \in S). Prove that...

Homework Statement Let S={v1=[1,0,0,0],v2=[4,0,0,0],v3=[0,1,0,0],v4=[2,-1,0,0],v5=[0,0,1,0]} Let W=spanS. Find a basis for W. What is dim(W)? Homework Equations The Attempt at a Solution i know that a basis is composed of linearly independent sets. This particular problem's...

If S\subseteq V and V is a vector space, then S is a vector space. Assume S isn't a vector space. Since S isn't a vector space, then V isn't a vector space; however, V is a vector space. By contradiction, S is a subspace. Correct?

Homework Statement Find the basis for the subspaces of R3 and R4 below. Homework Equations A) All vectors of the form (a,b,c), where a=0 B) All vectors of the form (a+c, a-b, b+c, -a+b) C) All vectors of the form (a,b,c), where a-b+5c=0 The Attempt at a Solution I honestly had...

hi guys, i have no idea of how to do the following question, could u give some ideas? Q:determine whether or not the given set forms a basis for the indicated subspace {(1,-1,0),(0,1,-1)}for the subspace of R^3 consisting of all (x,y,z) such that x+y+z=0 how should i start? i know the...

Homework Statement Hi, i don't know if you can help me but i am currently studying for my finals and i have come across a question which i am very confused about. i have looked it up in books but there seems to be no answer there. the question is Write down a vector of length 1 that is...

Today I was reading in a probabilities textbook that the probability of the union of two events is: p(E_1 \cup E_2) = p(E_1) + p(E_2) - p(E_1 \cap E_2) and reminded me of the similarity with the dimension of the union of two subspaces of a vector space: dim(V_1 \cup V_2) = dim(V_1) +...

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...