Basis Definition and 1000 Threads

The problem is attached. I am instructed to find a basis for the nullspace of T.A basis for a 2x2 matrix is 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1Applying the transformation to each of these gives 0 0 0 0 0 2 0 0 0 0 -2 0 0 0 0 0 respectively. Now this is where I get stuck. How do I find a...

The problem is attached. The problem is "find a basis for the range of the linear transformation T." p(x) are polynomials of at most degree 3. R(T)={p''+p'+p(0) of atmost degree 2} This is pretty much as far as I got. I'm not sure how to do the rest. I'm thinking of picking a...

T: P2 → R (the 2 is supposed to be a subscript) The P is supposed to be some weird looking P denoting that it is a polynomial of degree 2. T (p(x)) = p(0) Find a basis for nullspace of linear transformation T.The answer is {x, x^2} I want to make sure I'm interpreting this correctly. It...

This thread is spawned from an earlier one https://www.physicsforums.com/showthread.php?t=647147&page=7 For the stationary ( ie comoving ) frame in the Schwarzschild spacetime the co-basis of the frame field is s_0= \sqrt{\frac{r-2m}{r}}dt,\ \ s_1=\sqrt{\frac{r}{r-2m}}\ dr,\ \ s_2=r\...

Suppose that S = {v1, v2, v3} is a basis for a vector space V. a. Determine whether the set T = {v1, v1 + v2, v1 + v2 + v3} is a basis for V. b. Determine whether the set W = {−v2 + v3, 3v1 + 2v2 + v3, v1 − v2 + 2v3} is a basis for V. I must check if they're linearly independent...

The problem is attached. I'm having problems with parts a and c, well maybe not part a (probably just need to check if I did this part right. I'm just not sure if I'm wording part a right. Anyways for part a I must prove it's a subspace so I must satisfy 3 conditions: 1) 0 is in S 2) if U and...

I know how to do the problem, just put the 4 vectors in matrix form and find for what values of k is the detminant =0. the answer is then that k can't equal the value that was found. Is there a easier way to do this? My method involves finding the determinant using the expansion method...

Show that if S = {v1, v2, . . . , vn} is a basis for R
n and A is an n × n invertible matrix, then S' = {Av1,Av2, . . .,Avn} is also a basis. I need to show that: 1) Av1, Av2,...Avn are linearly independent 2) span(S)=Rn I'm having some problems with this. I see that S'=AS (duh)...

I'm not sure how to start this problem. All i know is a diagonal matrix consists of all 0 elements except along the main diagonal. But how do I even find a basis for this?

Hello I have this problem, I find it difficult, any hints will be appreciated... Two subspaces are given (W1 and W2) from the vector space of matrices from order 2x2. W1 is the subspace of upper triangular matrices W2 is the subspace spanned by...

According to my professor, there exist infinite dimensional vector spaces without a basis, and he asked us to find one. But isn't this impossible? The definition of a dimension is the number of elements in the basis of the vector space. So if the space is infinite-dimensional, then the basis...

Homework Statement The problem is Exercise 2 in the picture http://postimage.org/image/3ou3x1sh7/ Homework Equations The hint says: can you express and three-dimensional vector in terms of just two linearly independent vectors? The Attempt at a Solution I have no idea where...

Homework Statement Show that B = {x2 −1,2x2 +x−3,3x2 +x} is a basis for P2(R). Show that the differentiation map D : P2(R) → P2(R) is a linear transformation. Finally, find the following matrix representations of D: DSt←St, DSt←B and DB←B. Homework Equations The Attempt at a...

Hi! There is a concept I don't understand and would love to have is cleared... What is the meaning of a lattice with a basis? What do I need it for? Say I have a honeycomb structure. (fig 1) and a basis as mentioned there (did I understand it right? is it the basis? ) why does it become...

When you have a matrix like: 3 1 0 1 The RREF is 1 0 0 1 the identity matrix. Is the basis always [0 0]^t?

Suppose that T1: V → V and T2: V → V are linear operators and {v1, . . . , vn} is a basis for V . If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n, show that T1(v) = T2(v) for all v in V . I don't understand this question. They said If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n...

Homework Statement The set K of 2 × 2 real matrices of the form [a b, -b a] form a field with the usual operations. It should be clear to you that M22(R) is a vector space over K. What is the dimension of M22(R) over K? Justify your answer by displaying a basis and proving that the set...

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...

Hello Everyone, Can anyone tell me how to do a transformation from atomic basis to molecular basis incase of 2-electron integrals for the case of Gaussian type Orbitals? Like in case of One electron integrals it is known that a transformation from the Gaussian type orbitals to their...

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 &...