Are All Bases Sets of Orthogonal Vectors?

pr0me7heu2
Messages
13
Reaction score
2
Would all bases be sets of orthogonal (but not necessarily orthonormal) vectors?
 
Physics news on Phys.org
No. Try to think of a basis for R^2 consisting of nonorthogonal vectors -- there are plenty.
 
For example, {\vec{i}, \vec{i}+ \vec{j} } is a basis for R2 and they are not orthogonal (with the "usual" inner product). It happens to be easier to to find components in an orthonormal basis.

In any case, "orthogonal" as well as "orthonormal" depend upon an innerproduct defined on the vector space. Given any basis it is always possible to define an innerproduct in which that basis is orthonormal.
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I Non orthogonal basis and the lines of its coordinate grid
Replies
9
Views
3K
I Standard basis and other bases...
Replies
9
Views
3K
I Why Use Gram-Schmidt to Make a Unitary Matrix?
Replies
14
Views
4K
I What is the difference between a complete basis and an overcomplete dictionary?
Replies
5
Views
2K
I Vector subspace and basis vectors in the context of data science
Replies
6
Views
3K
I Basis of a Subspace of a Vector Space
Replies
8
Views
3K
I Finding the orthogonal projection of a vector without an orthogonal basis
Replies
3
Views
4K
MHB Finding a Basis for a Linear Subspace Orthogonal to a Given Point P in R^3
Replies
2
Views
1K
I Proof that if T is Hermitian, eigenvectors form an orthonormal basis
Replies
3
Views
3K
I Bases, operators and eigenvectors
Replies
7
Views
2K

Hot Threads

Recent Insights

Back
Top