Basis Definition and 1000 Threads

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...

Homework Statement Find a basis for the solution space of the homogeneous systems of linear equations AX=0 Homework Equations Let A=1 2 3 4 5 6 6 6 5 4 3 3 1 2 3 4 5 6 and X= x y z...

Hi, I'm having trouble understanding how people can make calculations using the partial derivatives as basis vectors on a manifold. Are you allowed to specify a scalar field on which they can operate? eg. in GR, can you define f(x,y,z,t) = x + y + z + t, in order to recover the Cartesian...

Homework Statement Consider a linear transformation L from Rm to Rn. Show that there is an orthonormal basis {v1,...,vm} of Rm to Rn such that the vectors {L(v1),...,L(vm)} are orthogonal. Note that some of the vectors L(vi) may be zero. HINT: Consider an orthonormal basis {v1,...,vm} for...

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 Given one known orthonormal basis S in terms of the standard basis U, how would I express a third basis T in terms of U when I know its representation in S? For example, U consists of <1,0,0> <0,1,0> <0,0,1> And S consists of (for example) <0.36, 0.48, -0.8>...

Hi, I understand that the maximum packing fraction for a particular atomic structure can be calculated assuming the nearset neighbours are touching but my question is how can the maximum packing fraction be calculated for a basis containing two different types of atoms? Thanks, James

Let S be the subspace of R^3 spanned by x=(1,-1,1)^T. Find a basis for the orthogonal complement of S. I don't even know where to start... I would appreciate your help!

Hi, I'm reading Shankar's Principles of QM and I find it not very clear on how exactly should I change basis of operator. I know how to change basis of a vector so can I treat the columns of operator matrix as vectors and change them? Or is it something else?

Homework Statement The set B = {-4-x^2, -8+4x-2x^2, -14+12x-4x^2} is a basis for P2. Find the coordinates of p(x) = (-2 +0x -x^2) relative to this basis. Homework Equations n/a The Attempt at a Solution so the set would be in a matrix like this: |-4 0 -1|...

I am having no luck understanding how to find the basis of a field adjoined with an element. For example Q(sqrt(2)+sqrt(3)) I know that if i take a=sqrt(2)+sqrt(3) that i can find a polynomial (1/4)x^4 - (5/2)x^2 + 1/4 that when evaluated at a is equal to zero. So, from that I know the...

We want to find a basis for W and W_perpendicular for W=span({(i,0,1)}) =Span({w1}) in C^3 a vector x =(a,b,c) in W_perp satisfies <w1,x> = 0 => ai + c = 0 => c=-ai Thus a vector x in W_perp is x = (a,b,-ai) So an orthonormal basis in W would be simply w1/norm(w1) ...but the norm(w1)=0 (i^2 +...

Homework Statement Recall that the matrix for T: R^{2} \rightarrow R^{2} defined by rotation through an angle \theta with respect to the standard basis for R^{2} is \[A =\begin{array}{cc}cos \theta & -sin \theta \\sin \theta & cos\theta \\\end{array}\]\right] a) What is the matrix of T...

Hi. Thanks for the help. Homework Statement Find a basis for the set of polynomials in P3 with P'(1)=0 and P''(2)=0. Homework Equations P' is the first derivative, P'' is the second derivative. The Attempt at a Solution The general form of a polynomial in P3 is ax^3+bx^2+cx+d...

Homework Statement My notes has the following statement, but I seem to have forgotten to write down the conclusion of the statement before my professor erased it from the board. "Any vector space V there will be a basis except for 1 type of space: " Any ideas as to what that 1 type of...

Homework Statement Define a non-zero linear functional y on C^2 such that if x1=(1,1,1) and x2=(1,1,-1), then [x1,y]=[x2,y]=0. Homework Equations N/A The Attempt at a Solution Le X = {x1,x2,...,xn} be a basis in C3 whose first m elements are in M (and form a basis in M). Let X' be...

Are there any sets of basis functions that are particularly useful for Maxwell's equations? I was thinking about Fourier just because it is the first basis I always think of, but I don't know that it would actually be a convenient basis. For example, I don't know that curl or divergence would...

Homework Statement Find the matrix of f relative to Alpha' and Beta'. Alpha' = [(1,0,0), (1,1,0), (2,-1,1)] Beta' = [(-13,9), (10,-7)] The question originally reads that f is a bilinear form. I've found a (correct according to answer key) matrix A that is 3 -4 4 -5 -1 2...

Hey guys! I am having a major brain problem today, with this problem. L is a linear transform that maps L:P4\rightarrowP4 As such that (a1t3+a2t2+a3t+a4 = (a1-a2)t3+(a3-a4)t. I am trying to find the basis for the kernel and range. I know that the standard basis for P4 is...

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 Consider the vector \vec A whose origin is \vec r. 1)Express the vector \vec A in a basis of Cartesian coordinates, cylindrical and spherical ones. 2)Repeat part 1) if the origin is \vec r + \vec r_0. Homework Equations None given. The Attempt at a Solution 1)In...

Hello. My question is: does the norm on a space depend on the choice of basis for that space? Here's my line of reasoning: If the set of vectors V = \left\{ v_1,v_2\right\} is a basis for the 2-dimensional vector space X and x \in X, then let \left(x\right)_V = \left( c_1,c_2\right)...

If S is subspace of R6x6 consisting of all lower triangular matrices, what is the dimension of S? Does anyone know the properties about dimension of lower triangular matrices?

My mind is shot. Let S be a subspace of P[SUB]3[SUB] consisting of all polynomials of the form ax2+bx+2a+3b. Find a basis for S. I am not sure where to start.

Homework Statement Let T be the linear transformation of R5 into R3 that has the matrix A = 1 3 2 0 -1 2 6 4 6 4 1 3 2 2 1 relative to the bases [(1,1,1,1,1), (1,1,1,1,0), (1,1,0,0,0), (1,0,0,0,0), (0,0,0,0,1)] of R5 and [(1,1,1), (0,1,0), (1,0,0)] of R3. Find a basis for the range...

Hi guys 1) We are looking at a Hamiltonian H. I make a rotation in Hilbert space by the transformation {\cal H} = \mathbf a^\dagger\mathsf H \mathbf a = \mathbf a^\dagger \mathsf U\mathsf U^\dagger\mathsf H \mathsf U\mathsf U^\dagger\mathbf a = \mathbf b^\dagger...

I remember some of my linear algebra from my studies but can't wrap my head around this one. Homework Statement Say my solution to a DE is "f(x)" (happens to be bessel's equation), and it contains a constant variable "d" in the argument of the bessel's functions (i.,e. J(d*x) and Y(d*x)). So...

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...

Homework Statement Let U and V be vector spaces of dimensions of n and m over K and let Hom(subscriptK)(U,V) be the vector space over K of all linear maps from U to V. Find the dimension and describe a basis of Hom(subscriptK)(U,V). (You may find it helpful to use the correspondence with mxn...

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 I have a set of Vector v_1,v_2,v_3,v_4 in \mathbb{R}^4 and need to show that E = v_1,v_2,v_3,v_4 is an ordered basis for \mathbb{R}^4 The Attempt at a Solution I know that for this being the case v = c_1 \cdot v_1 + \cdots + c_4\cdot v_4 where v \in...

Homework Statement Let L be the line in R^3 spanned by v1=(1,1,1) Find a basis (v2,v3) for the plane perpendicular to L, and verify that B=(v1,v2,v3) is a basis for R^3. Homework Equations The Attempt at a Solution I know that if two vectors are perpendicular or orthogonal that...

Homework Statement The matrix A =(1,2,3;4 5 6) defines a linear transformation T: R^3-->R^2 . Find the transformation matrix for T with respect to the basis (1,0,1),(0,2,0),(-1,0,1) for R^3 and the basis (0,1),(1,0) for R^2. Homework Equations - The Attempt at a Solution I have no...

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 What is a basis for the space of 2 x 2 matrices. The Attempt at a Solution I don't understand how to this at all. Is the 2x2 identity matrix abasis for 2x2 matrices? Because it's linearly independent and spanning the space? Can anyone explain?

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...

Linear Algebra - Change of basis matrices and RREF question what in the world?? Homework Statement Suppose the linear transformation T: P3 -> P2, over R has the matrix A = \begin{bmatrix}1&2&0&0\\0&1&2&1\\1&1&1&1 \end{bmatrix}...

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Homework Statement erm, I just want to know, what is the basis for a zero vector space? Homework Equations The Attempt at a Solution is it the zero vector itself? but if that's the case, then the constant alpha could be anything other than zero, which means the zero vector is not...

Homework Statement Let A = E4 in R4 (standard basis) and B = {x^2, x, 1} in P2 over R. If T is the linear transformation that is represented by \begin{bmatrix}1 & 1 & 0 & 1\\0 & 0 & 1 & -1\\1 & 1 & 0 & 1 \end{bmatrix} relative to A and B, find...

Homework Statement Let A = {(1, 1), (2,0)} and B = {(0, 2), (2, 1)} in R2. a) Find [u]A (u with respect to A) if [u]B = [3, -2]. Homework Equations The Attempt at a Solution I tried to find [I]AB (transition matrix from B to A), then apply to [u]B, but couldn't represent (2, 1)...

I'm teaching myself quantum mechanics and am learning about bra-ket notation. There is a particular operator used, the ket-bra (e.g. |X><X|). To understand it, I'm trying to come up with an orthonormal basis for C^2 as a simple case (i.e., the 2-dimensional vector space over the field of complex...

Hi, I have an assignment due and I have done most of the questions there are just a couple things I have left, if someone can help that would be amazing :) 1. In this problem we suppose that F is a field, A is an m by n matrix over F and that W is a subspace of Fm. (a) Show that U =...

Homework Statement Show that the functions po(t)=1, p1(t)=t, p2(t)=1/2(3t2-1), and p3(t)=(3/2)*[(5/3)t3-t) also form a basis for the vector space P3(R) ... "R" meaning all real numbers Homework Equations I know these polynomials are the first four Legendre polynomials The Attempt...

Homework Statement A is a mxn matrix, and P is an invertible nxn matrix. So I want to prove that the bases of null(A) and null(AP) have the same number of elements. Homework Equations The Attempt at a Solution I was going to start off by assuming that {X1, X2, ... Xm} is a...

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...

Homework Statement Among all independent vector sets in a vector space U, let M = {v1, v2, ... vp} be an independent set. p is as large as it can get. Show that U is a basis of M. Homework Equations The Attempt at a Solution If U is a basis of M then U is an independent set (we...

Homework Statement Find the dimnesion and a basis of vector space V Homework Equations V is the set of all vectors (a,b,c) in R^3 with a+2b-4c=0 The Attempt at a Solution (4c-2b,b,c) = b(-2,1,0) + c(4,0,1) so {(-2,1,0),(4,0,1)} is the basis of the SUBSPACE of V right? how do I...

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...

Helo all, I have a very simple question about basis Four-Vectors and Components. In Hartle's book, Gravity, he uses the following equation to show the components of the 4-vector, a a =a^t{}e(sub t) + a^x{}e(sub x) + a^y{}e(sub y) + a^z{}e(sub z) Sorry for the half LaTex half something...