Basis Definition and 1000 Threads

I'm guessing this is the right forum to post in. Ammonia shares many properties with water. It is polar, it is amphoteric, it reacts with itself to form its acid and base conjugates NH4+ and NH2-. Just as water is our basis of life, could another species use ammonia like we use water...

I was explaining basis vectors to my brother, I said that in quantum mechanics that when you have a number of dimensions, each dimension being an eigenket in vector space, that every dimension is independent of all the other basis vectors. It is however interesting to think that if this is the...

Hello, I am new to the forums and I hope this fundamental topic has not been previously treated, as these forums don't seem to have a search function. I am studying general relativity using S. Carroll's book (Geometry and Spacetime) and I am having a fundamental problem with basis vectors under...

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)?

What is the so called "orthogonal operator basis"? What is the so called "orthogonal operator basis"?

I want to write a program that can do algebraic transformations and mathematical deduction for me. It's not meant to do anything by itself, but rather check the transformations that I do myself for validity. The set of rule I will specify in advance. I want to capture all/most of the...

As far as I can make out, the hallowed principle of conservation of mass-energy (modulo quantum fluctuations) lies on four principles: (1) from the assumption that every effect has a cause (again, modulo the leeway given by Heisenberg), so if there is no mechanism for creation or destruction of...

Homework Statement It's not a homework problem. I'm reading my textbook (Sakurai's Modern QM), and I'm not sure about a step (eq 3.6.6 through 3.6.8). Here it is: We start with a wave function that's been rotated: \langle x' + y' \delta \phi, y' - x' \delta \phi, z' | \alpha \rangle Now...

I have my linear algebra exam coming up but I missed the class on bases. Can anyone show me how this is solved? 2. Consider the subspace U of R4 defined by U = span{(−1, 1, 0, 2), (1, 0, 0, 1), (2,−1, 1,−1), (0, 1, 0, 3)} • Find a basis of U. • Find a basis of the orthogonal complement U.

A few mornings ago, I helped a young woman with her car (dead battery), which made me late for work. Which brought some negative consequences to me. Yet I would do the same thing again just to help. With no benefit to me, but with negative consequences, I felt good for helping someone in...

Could someone please tell me what the preferred basis problem is with regards to the Everett/many worlds interpretation? As I understand it, it refers to the basis which is needed to make macroscopic objects determinate in all worlds. But what does this mean? Does basis refer to an observable...

Hey all! I am working on this and got confused. Any help at all would be much appreciated! Determine the kernel and range of the transformation T and find a basis for each: T(x,y,z)=(x,y,z) from R3 to R2. I have found the kernel to be the set {(r, -r, 0)}. Range is R2. I"m not sure how...

Homework Statement Let S be the form of (a, b,c ,d )in R4, given a not equal to 0. Find the basis that is subset of S.Homework Equations The Attempt at a Solution I got a(1,0,0,0), b(0,1,0,0), c(0,0,1,0), d(0,0,0,1) as basis. a not = 0 But i wasn't sure what the significances of a not = to 0...

Inspired by the Dirac bra-ket notation I came to think that an ordinary Euclidean vector must be expressible without reference to a basis. But if I specify the length and angle of a vector, I have to refer this angle to some particular direction. Isn't this the same as choosing a basis? Edit...

In short: does every vector space have a "standard" basis in the sense as it is usually defined i.e. the set {(0,1),(1,0)} for R2? And another example is the standard basis for P3 which is the set {1,t,t2}. But for more abstract or odd vector spaces such as the space of linear transformations...

Homework Statement Says, The set W = {(x,y,z,w) : x+z=0, 2y+w=0} is a subspace of R^4. Find a basis for W, and state the dimension. The Attempt at a Solution What I did: W= {(-z,-w/2,z,w): z,w are in R} = {z(-1,0,1,0) + w(0,-1/2, 0, 1)} = span {(-1,0,1,0), (0,-1/2,0,1)} (-1,0,1,0)...

Hi! I am working on the following problem: If a matrix is antisymmetric (thus A^T = -A), show that P = {A \in R | A is antisymmetric} is a subset of Rnxn. Also, find the dimension of P. So far, I've proven that P is a subset of R and I am guessing that, in order for me to find the...

Homework Statement In the xy-plane, sketch the coordinate system [ a; b] corresponding to the basis { (1, 1 ) , (1, -1) } by drawing the lines a = 0, \pm1 and b = 0, \pm1. What point in the xy-plan corresponds to a = 1, b = 2?Homework Equations Not sure of any in this caseThe Attempt at a...

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...

Homework Statement Find the matrix of the linear operator with respect to the given basis B. D: P2 -> P2 defined by D(ax2 + bx + c) = 2ax+b, B = { 3x2+2x+1, x2-2x, x2+x+1 }Homework Equations None. The Attempt at a Solution I set the basis B = { (3,2,1), (1,-2,0), (1,1,1) } based on the...

Dear Physics Forum, I have a question about quantum mechanics. I know that the solutions of a Hamiltonian will form a complete basis. However, for a case of a finite well, in the region of bound states (E<0), the number of eigenfunctions is finite, I wonder that they are enough to form a...

We have three orthonormal vectors \vec i_1 , \vec i_2, \vec i_3 , and we know which are the components of an arbitrary vector \vec A in this base, explicitly: \vec A = (\vec A \bullet \vec i_1) \vec i_1 + (\vec A \bullet \vec i_2) \vec i_2 + (\vec A \bullet \vec i_3) \vec i_3...

If A is an invertible matrix and vectors (v1,v2,...,vn) is a basis for Rn, prove that (Av1,Av2,...,Avn) is also a basis for Rn.

Homework Statement show that \left(\begin{array}{cc}2 & -1\\-1 & 1\end{array}\right) forms a basis for R^2Homework Equations The Attempt at a Solution ok...my instructor said he wants me to show that they are linearly independant and to show that they span to form a basis...not just by a...

Homework Statement Let {e1,e2,e3} be a basis for the vector space V, and T:V \rightarrow V a linear transformation. let f1 ;= e1 f2;=e1+e2 f3;=e1+e2+e3 Find the Matrix B of T with respect to {f1,f2,f3} given that the matrix with respect to {e1,e2,e3} is \[ \left(...

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?

Homework Statement 1. Which of the following is not a linear transformation from 3 to 3? a. T(x, y, z) = (x, 2y, 3x - y) b. T(x, y, z) = (x - y, 0, y - z) c. T(x, y, z) = (0, 0, 0) d. T(x, y, z) = (1, x, z) e. T(x, y, z) = (2x, 2y, 5z) 2. Which of the following...

Homework Statement I have used the gram schmidt process to find an orthogonal basis for {1,t,t^2} which is (1,x,x^2 - \frac{2}{3}) How to i normalize these Homework Equations e_1=\frac{u_1}{|u_1|} The Attempt at a Solution...

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...

If T={Bi} Bi are the all matrix of rank n. So,T is a matrix space(right?). How to calculate the basis of T? are the basis of T also some matrix? Thank you!

this is apparently "really simple", but I just don't know how to do it from the examples I have and I feel like a moron... what's the basis for the nullspace of this matrix [ 2 3 1] [ 5 2 1] [ 1 7 2] [ 6 -2 0]

Homework Statement Let T : M2,2, --> M2,1 be the linear transformation given by T ([a b; c d]) = [a-2b ; c-2d] Fix bases B = { [1 0 ; 0 0], [ 0 1 ; 0 0], [0 0 ; 1 0], [0 0 ; 0 1]} and C = { [1 ; 0], [0 ; 1]} for M2,2, and M2,1 respectively. (a) Find the matrix [T]C,B of T with...

Hi everyone, I am working on the following problem. Suppose the set of vectors X1,..,Xk is a basis for linear space V1. Suppose the set of vectors Y1,..,Yk is also a basis for linear space V1. Clearly the linear space spanned by the Xs equals the linear space spanned by the Ys. Set X=[X1: X2...

I am in a linear algebra and differential equations course and have recently been learning how to find a basis for a nullspace, row space, or column space. However, I am EXTREMELY confused by a solution to a question in my textbook. The question asks to find the basis for the null space of a...

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...

Normally, if you have an orthonormal basis for a space, you can just apply your metric tensor to get your dual basis, since for an orthonormal basis all the dot products between the base vectors will boil down to a Kronecker delta. However, in Minkowski space, the dot product between a unit...

Homework Statement Give the basis and dimension of the set of all 2x2 complex symmetric matrices. Homework Equations The Attempt at a Solution I know that if the coefficients were real, then I could just have the basis \left( \begin{array}{cc} 1 & 0\\ 0 & 0 \end{array}...

Homework Statement Prove: if an n × n matrix A is orthogonal (column vectors are orthonormal), then the columns form an orthonormal basis for R^n. (with respect to the standard Euclidean inner product [= the dot product]). Homework Equations None. The Attempt at a Solution I...

Hi. I found in a book on quantum optics (Vogel, Welsch - Quantum Optics) and also in a lecture script the following statements. A System is composed of two subsystems (say A and B) which interact. So the total Hamiltionian is H_{AB}= H_A + H_B + H_{int}. Nontheless the states of H_{AB}...

How am i supposed to write eigenkets of an operator whose matrix is given to me given that the two ket vectors form an orthonormal basis .

Homework Statement Define T : R3x1 to R3x1 by T = (x1, x2,x3)T = (x1, x1+x2, x1+x2+x3)T 1 Show that T is a linear transformation 2 Find [T] the matrix of T relative to the standard basis. 3 Find the matrix [T]' relative to the basis B' = {(1,0,0)t, (1,1,0)t, (1,1,1)t 4 Find the...

Homework Statement Find an Basis for Image and Kernel of the matrix. \[ \left( \begin{array}{ccc} 2 & 1 & 3 \\ 0 & 2 & 5 \\ 1 & 1 & 1 \end{array} \right)\] Homework Equations The Attempt at a Solution To find the kernel I solve the equation Ax = 0 I put the matrix in row...

1. The problem statement Let W = {(x, y, z, t): x + y + 2z - t = 0} be a vector space under R^4. Find a basis of W over R. 2. The attempt at a solution To me I would think that the vector space itself could its own basis, but I know I'm probably way off. I also tried solving x = t - y...

Homework Statement I need to prove this formula, but I'm not sure how to prove it.[T]C = P(C<-B).[T]B.P(C<-B)-1 whereby B and C are bases in finite dimensional vector space V, and T is a linear transformation. Your help is greatly appreciated! Homework Equations T(x)=Ax [x]C=P(C<-B)[x]B...

Given V = {(x1; x2;….; xn) | Σni=1 xi=0} (sum of vectors is equal to zero) be a subspace of Rn. How can we find a basis of V such that for each vector {(x1; x2;….; xn) in the basis Σi=1n x2i=1 ( i.e. sum of squares is equal to 1).

Homework Statement The matrix A= 2 0 4 -2 0 -4 -1 0 -2 has one real eigenvalue. Find this eigenvalue and a basis of the eigenspace. Eigenvalue = Basis ( , , )T , ( , , )T Homework Equations The Attempt at a Solution I have found the...

Homework Statement Let {e1,e2,e3} be a basis for the vgector space V over the field F. Put f1 = -e1, f2 = e1+e2 and f3 =e1 + e3 Prove that {f1,f2,f3} is also a basis for V Homework Equations The Attempt at a Solution I made e1,e2,e3 be the unit bases. 1...

Hello, I was wondering if there has been any theoretical work regarding the thermodynamic justification of the Exclusion Principle. It is my view that all open systems proceed to the lowest free-energy state and all closed systems go to the highest entropy state. Of course, these states must...

Homework Statement "Prove that a set S of three vectors in V3 is a basis for V3 if and only if its linear span L(S) contains the three unit coordinate vectors i, j, k." Homework Equations I have the definitions of bases, linear independence, and linear spans. I have the theorems which...

hi, I have a quickon vector spaces. Say for example we have X = a1U1 + a2U2 ...anUn this can be written as X = sum of ( i=0 to n) ai Ui now how can I get and expression of ai in therms of X and Ui. do we use inner product to do this...ans someone please explain how to go...