Basis Definition and 1000 Threads

Homework Statement Determine the column vectors representing the states |+x> and |-x> using the states |+y> and |-y> as a basis. Homework Equations ? The Attempt at a Solution The hint my prof gave us was that since |+x> = 1/√2|+z> + 1/√2|-z> we can eliminate the states |+z> and...

So the title says everything. Let's assume R is a set equipped with vector addition the same way we add real numbers and has a scalar multiplication that the scalars come from the field Q. I believe the dimension of this vector space is infinite, and the reason is we have transcendental numbers...

Hello Forum, When we represent a vector X using an orthonormal basis, we express X as a linear combination of the basis vectors: x= a1 v1 + a2 v2 + a3 v3+ ... Each coefficient a_i is the dot product between x and each basis vector v_i. If the vector x is not a row (or column vector)...

Hi, I've been trying to prove that every vector space has a basis. So starting from the axioms of vector space I defined linear independence and span and then defined basis to be linear independent set that spans the space. I was trying to figure out a direct way to prove the existence of...

Homework Statement This is something I should know, but I keep getting mixed up when I try to think about it. A quantum state can be written as a superposition of basis states such as \left | n \right \rangle So let's say I have a particle in a potential with discrete energy levels...

In analogy to vector spaces, can we define a set of "basis functions" from which any continuous function can be written as a (possibly infinite) linear combination of the basis functions? I know the trigonometric functions 1, sin(nx), cos(nx) can be used for monotonic continuous functions...

I attached 2 problems. For problem #1. I want to make sure I'm on the right track, to find the span of Null(A), i need to put matrix A in RREF form. By doing so I get x1=-2t x2=-t x3=s x4=u (using u because I'm using t to denote transpose) where x1 to x4 is for each respective column...

Homework Statement Let A \in M_n(F) and v \in F^n. Let v, Av, A^2v, ... , A^{k-1}v be a basis, B, of V. Let T:V \rightarrow V be induced by multiplication by A:T(w) = Aw for w in V. Find [T]_B, the matrix of T with respect to B. Thanks in advance Homework Equations...

I am really confused about something. I know that if I have a vector space, then the dimension of that vector space is the number of elements in a basis for it. But this brings up some confusing issues for me. For example, if we are looking at the null space of a non-singular, square matrix...

Homework Statement Find a basis for the subspace S of vectors (A+B, A-B+2C, B, C) in R4 What is the dimension of S? The Attempt at a Solution Do I just plug in varying values for A B and C to create four vectors, and see if they are linearly independent? If they are then I've found...

1. What can be said of the dimension of the basis of the Reals over the Irrationals 2. Homework Equations 3. I believe the basis is infinite because any real number can be made out of the combination of irrational vectors multiplied by the same irrational coefficient to make any real number...

Find http://imageshack.us/a/img35/1637/lineal2.gif http://imageshack.us/a/img210/1370/lineal1.gif C^3 is the canonical base of ℝ^3, C^2 is the canonical base of ℝ^2 I tried: http://imageshack.us/a/img822/6274/lineal3.gif But I'm not sure if this is right, I made a...

Find a basis for and the dimension of the subspaces defined for each of the following sets of conditions: {p \in P3(R) | p(2) = p(-1) = 0 } { f\inSpan{ex, e2x, e3x} | f(0) = f'(0) = 0} Attempt: Having trouble getting started... So I think my issue is interpreting what those sets...

Hello! Im trying to read some mathematical physics and have problems with the understanding of vector fields. Th questions are regarding the explanations in the book "Geometrical methods of mathematical physics".. The author, Bernard Schutz, writes: "Given a coordinate system x^i, it is often...

Hi guys, Let's say I have a 6x6 matrix A whose Jordan form J has 3 Jordan blocks. It means that this matrix (matrix A, but I think that also the matrix J) has 3 linearly independent eigenvectors, I have no problem in finding them. I simply do (A-\lambda _i I)v_i=0 to get the eigenvectors v_i...

How do i go about this? Find a basis for the subspace W of R^5 given by... W = {x E R^5 : x . a = x . b = x . c = 0}, where a = (1, 0, 2, -1, -1), b = (2, 1, 1, 1, 0) and c = (4, 3, -1, 5, 2). Determine the dimension of W. (as usual, "x . a" denotes the dot (inner) product of the...

Greetings, I have just started studying manifolds, and have come across the idea that the basis vectors can be expressed as: e\mu = \partial/\partialx\mu. I tried to convince myself of this in 2D Cartesian coordinates using a pretty non-rigorous derivation (the idea being to get a...

Hi, I'm beginning to learn QM, and I've never seen any treatment of vector spaces with infinite bases. Countable case is quite digestible, but uncountable just flies over my head. Can anyone recommend me place where to learn this more advanced part of linear algebra, with focus on stuff...

I am pretty sure I want to go into engineering, but I am really curious as to what engineers do on a daily basis. I have this vague and somewhat childish idea that it is just a bunch of people in overalls tinkering with machine parts. That just shows how little I really know. I'm looking for...

Homework Statement A = \left( \begin{array}{ccc} 2 & 0 & -1 \\ 4 & 1 & -4 \\ 2 & 0 & -1 \end{array} \right) Find the eigenvalues and corresponding eigenvectors that form a basis over R3 Homework Equations The Attempt at a Solution OK so I've found the characteristic...

Homework Statement Greetings, I have been stuck with this problem for a while, I thought maybe someone could give me some advice about it. Thanks a lot in advance. If T is a linear transformation that goes from R^2 to R^2 given that T(v1)= -2v2 -v1 and T(v2)=3v2. and B =...

Homework Statement I'm having trouble understanding a concept in representation theory. I've been reading several texts on the application of rep. theory to quantum mechanics ("Group Theory and Quantum Mechanics" by Tinkham and "Group Theory and Its Application to Physical Problems" by...

Hi, I have a function on [0,\infty) which is represented as: \sum_{ \stackrel{ \Re( \alpha )\in\mathbb{Q}^+ }{ \Im( \alpha )\in\mathbb{Q} } }{\beta_\alpha e^{-\alpha t}} It seems like this must be a basis for the square integrable functions on [0,\infty) with exponential tails. Am I right...

Find a basis of the given span{[2,1,0,-1],[-1,1,1,1],[2,7,4,5]} So I got the RREF, and found the basis to be two rows of the RREF, which are [1,0,-1/3,0] and [0,1,2/3,1], but the answer is [2,1,0,-1],[-1,1,1,1] Where did I do wrong?

I am not sure -- a manifold is locally connected and has countable basis? There is an Exercise in a book as following : Given a Manifold M , if N is a sub-manifold , an V is open set then V \cap N is a countable collection of connected open sets . I am asking why he put this exercise...

I have normally introduced basis vectors by just stating independent vectors that span the space. This is perhaps not very inspirational. What is attractive way to introduce basis vectors? I am looking for a hook that students will find motivating. It needs to have an impact. Maybe a good...

What is attractive way to introduce basis vectors? I am looking for a hook that students will find motivating. It needs to have an impact. I have normally introduced it by just stating independent vectors that span the space.

I got trouble to understand the cyclic sum identity (the first Bianchi identity) of the Riemann curvature tensor: {R^\alpha}_{[ \beta \gamma \delta ]}=0 or equivalently, {R^\alpha}_{\beta \gamma \delta}+{R^\alpha}_{\gamma \delta \beta}+{R^\alpha}_{\delta \beta \gamma}=0. I can understand the...

Homework Statement https://dl.dropbox.com/u/4788304/Screen%20shot%202012-07-08%20at%2002.53.44.JPG This is the solution of Problem A.15 in Griffiths' Quantum Mechanics. Tx is the rotation matrix about x-axis for theta degrees; while Ty is the rotation matrix about y-axis for theta degrees...

Do you people know any decent and simple (no fancy math) material for the atomic physics around diamagnetism, paramagnetism and ferromagnetism? Topics like wave-particle duality, energy levels, solids, why metals are conductors, relation between magnetic moment and effective mass of...

Since it is based on the kinetic energy less the potential energy, what does the Lagrangian actually represent? Is there some intuitive way to understand why it is defined so and why it is such a fruitful concept using the principle of least action?

Hi all, I'm having trouble finding jordan basis for matrix A, e.g. the P matrix of: J=P^{-1}AP Given A = \begin{pmatrix} 4 & 1 & 1 & 1 \\ -1 & 2 & -1 & -1 \\ 6 & 1 & -1 & 1 \\ -6 & -1 & 4 & 2 \end{pmatrix} I found Jordan form to be: J = \begin{pmatrix} -2 & & & \\ & 3 & 1 & \\ & & 3 &...

{(a_i)_j} is the dual basis to the basis {(e_i)_j} I want to show that ((a_i)_1) \wedge (a_i)_2 \wedge... \wedge (a_i)_n ((e_i)_1,(e_i)_2,...,(e_i)_n) = 1 this is exercise 4.1(a) from Spivak. So my approach was: \BigWedge_ L=1^k (a_i)_L ((e_i)_1,...,(e_i)_n) = k! Alt(\BigCross_L=1^k...

Hello, I was wondering if the pseudoinverse can be considered a change of basis? If an m x n matrix with m < n and rank m and you wish to solve the system Ax = b, the solution would hold an infinite number of solutions; hence you form the pseudoinverse by A^T(A*A^T)^-1 and solve for x to...

Homework Statement Find a kernel and image basis of the linear transformation having: \displaystyle T:{{\mathbb{R}}^{3}}\to {{\mathbb{R}}^{3}} so that \displaystyle _{B}{{\left( T \right)}_{B}}=\left( \begin{matrix} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 0 & 0 & 0 \\ \end{matrix} \right)...

If V is a 3-dimensional Lie algebra with basis vectors E,F,G with Lie bracket relations [E,F]=G, [E,G]=0, [F,G]=0 and V' is the Lie algebra consisting of all 3x3 strictly upper triangular matrices with complex entries then would you say the following 2 mappings (isomorphisms) are different? I...

Hi there! A Hilbert space E is spanned by a set S if E is generated by the element of S. It is well known that in the finite dimensional case that S spans E and S is linearly independent set iff the set S form a basis for E. The question is that true for the infinite dimensional...

Homework Statement Let W = \begin{cases} \begin{pmatrix}x\\y\\z\\w\end{pmatrix} \in R^4 | w + 2x + 2y + 4z = 0 \end{cases} A)Find basis for W. B)Find basis for W^{\perp} C)Use parts (A) and (B) to find an orthogonal basis for R^4 with respect to the Euclidean inner product. Homework...

If we have any two orthonormal vectors A and B in R^2 and we wish to describe the circle they create under rigid rotation (i.e. they rotate at a fixed point and their length is preserved), how can we describe any point along this (unit) circle using a linear combination of A and B? I was...

Homework Statement Homework Equations ... The Attempt at a Solution Can someone just point me how to approach this? Do we take a random second degree polynomial and input 2x + 3 instead of x, then find the constants (eg. denoted by a , b , c) by putting the new equation equal to...

I've been working on this Linear Algebra problem for a while: Let F be a field, V a vector space over F with basis \mathcal{B}=\{b_i\mid i\in I\}. Let S be a subspace of V, and let \{B_1, \dotsc, B_k\} be a partition of \mathcal{B}. Suppose that S\cap \langle B_i\rangle\neq \{0\} for all i...

Homework Statement V = {p(x) belongs to P3 such that p'(1) + p'(-1) = 0} Homework Equations ... The Attempt at a Solution Okay, so finding the first derivative of p(x) = ax^3 + bx^2 + cx + d and plugging in the values 1 and -1 (to find p'(1) and p'(-1)), we get c = -3a. Does this make the...

Homework Statement $$ \begin{pmatrix} -1&3&0\\ 2&0&-1\\ 0&-6&1 \end{pmatrix} $$ Finding the ImT basis of this The Attempt at a Solution I got it down to $$ \begin{pmatrix} 1&0&-1/2\\ 0&1&1/6\\ 0&0&1 \end{pmatrix} $$ I know that by the principle of having pivots as the only non-zero...

Homework Statement The problem along with its solution is attached as Problem 1-2.jpg. Homework Equations Norm of a vector. The Attempt at a Solution Starting from the final answer of the solution, sqrt((-0.625)^2 + (0.333)^2) == 0.708176532 != 1. Did the book do something wrong? I ask...

I was wondering whether we can use row as well as column operations to reduce a matrix to find column space? Or do we only have to perform row operations to reduce matrix in case of row space and column operations to find column space?

Homework Statement w1 = 2 1 2 w2 = 1 -2 -3 w3 = 5 0 1 for R3 Homework Equations The Attempt at a Solution The books says the above is not a basis, why not? There are no free variables, none of the vectors are multiples of the other, they are linearly independent and the...

Im having trouble under stand the relationships between determinats, span, basis. Given a 3x3 matrix on R3 vector space. * If determinat is 0, it is linearly dependent, will NOT span R3, is NOT a basis of R3. , If determinant is non-zero, its linearly independent, will span R3, is a basis of...

From what I understand, a basis is essentially a subset of a vector space over a given field. Now what I'm not so sure of is the linearly independence part. If the basis has two linearly independent vectors, then than means they aren't collinear: rather, they wouldn't have the same slope...