Orthonormal basis Definition and 68 Threads

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...

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 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...

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 .

please help in this problem what are these basis and what are there there properties.how i can i put there values to solve my problems. ˆB  = ( 1 2  + +)er er  + ( 1 2  − +)e e  + (× + !)er e  + (× − !)e er  . (4.13) where eµ  are co-frame basis satisfying...

Homework Statement True/False: The set of vectors B={(-1,-1,1,1),(1,0,0,0),(0,1,0,0),(-1,-1,1,-1)} is an orthonormal basis for Euclidean 4-space \mathbb{R}^4. Homework Equations NoneThe Attempt at a Solution I said false because \langle (-1,-1,1,1),(-1,-1,1,1) \rangle =2\ne1, which shows...

Homework Statement Hey guys. http://img39.imageshack.us/img39/2345/27760913.jpg I need to show that these wave functions are orthonormal. I'm a bit confuse, what's i and what's j? I mean, do I need to take both of the functions, put them in the integral and to show that the result...

Homework Statement Let R^3 have the inner product <u, v> = u1v1 + 2u2v2 + 3u3v3. Use the Gram-Schmidt process to convert u1=(1,1,1), u2 = (1,1,0), u3 = (1,0,0) into a normal orthonormal basis Homework Equations I know the process for the orthonomoral converasion. I have no problem...

i got these vectors which are othronormal v1 (1/2,-1/2,1/2,-1/2) v2 (-1/2,1/2,1/2,-1/2) i need to compete them into orthonormal basis i did row reduction on them and added these independant vectors to the group v3(1,0,0,0) v4(0,1,0,0) now all four vectors are independant...

Homework Statement How can I find the orthonormal basis of four vectors? The vectors are: (0, 3, 0, 4), (4, 0, 3, 0), (4, 3, 3, 4) and (4, -3, 3, -4). The Attempt at a Solution I am not sure, whether I should use Gram-Schmidt process or the process of finding eigenvalues, eigenvectors and...

Homework Statement Find an orthonormal basis for the subspace of R^3 consisting of all vectors (a, b, c) such that a + b + c = 0. Homework Equations The Attempt at a Solution I know how to find an orthonormal basis just for R^3 by taking the standard basis vectors (1, 0, 0), (0...

Homework Statement My problem is I am getting a different answer than what MATLAB is giving me and I cannot determine why. Plz advise. Find an orthonormal basis of eigenvectors for matrix A= [3 2; 2 1] (using MATLAB notation- I couldn't figure out how to put in proper matrix notation)...

I hope this is the forum to ask this question. We all know that the eigenvectors of a Hermitian operator form an orthonormal basis. But is the opposite true as well. Are the vectors of an orthonormal basis always the eigenvectors of some Hermitian operator? Or do we need added restrictions to...

If A*A=AA* than A is a normal matrix. But does A must also have an orthonormal basis?

I need to find the Orthonormal Basis of this plane: x - 4y -z = 0 I know the result will be the span of two vectors but I'm not sure where to start. Any hints? Thanks, Gab

My other problem is: Consider now the space of 2x2 complex matrices. Show that the Pauli Matrices |I>= 1 0 0 1 |sigma x>= 0 1 1 0 |sigma y>= 0 -i i 0 |sigma z>= 1 0 0 -1 form an orthonormal basis for this space...

Hey there I'm working on questions for a sample review for finals I'm stuck on these three I think I'm starting to confuse all the different theorem, I'm so lost please help 1) Find the coordinate vector of the polynomial p(x)=1+x+x^2 relative to the following basis of P2: p1=1+x...

I'm wanting to form an orthonormal basis from two non-parallel vectors. a = \left(\begin{array}{cc}3 & 4\end{array}\right) b = \left(\begin{array}{cc}2 & -6\end{array}\right) Could someone please walk me through the calculations needed? Much appreciated.