Find a basis for the space of 2x2 symmetric matrices

ryanclarkeatm
Messages
2
Reaction score
0
a)Find a basis for the space of 2x2 symmetric matrices. Prove that your answer is indeed a basis.
b)Find the dimension of the space of n x n symmetric matrices. Justify your answer.
 
Physics news on Phys.org
The space of 2x2 matrices is in general isomorphic to a very familiar space. Think about the way addition of matrices and scalar multiplication work, and you should figure this out (and if you think about this for a while, you might realize a more general property about finite vector spaces over a field). From there, you should realize that the symmetric matrices are a subspace. If you look at some examples of 2x2 symmetric matrices, you should see the pattern.
 
A general 2 by 2 symmetric matrix is of the form
\begin{bmatrix}a & b \\ b & c \end{bmatrix}
you should be able to get a basis and the dimension from that.
 

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 Basis of 2x2 matrices with real entries
Replies
11
Views
5K
MHB Proving Properties of 2x2 Matrices
Replies
5
Views
2K
I Consistent matrix index notation when dealing with change of basis
Replies
12
Views
2K
I Congruence for Symmetric and non-Symmetric Matrices for Quadratic Form
Replies
7
Views
2K
I Transform a 2x2 matrix into an anti-symmetric matrix
Replies
5
Views
2K
Dimension of all 2x2 symmetric matrices?
Replies
3
Views
10K
MHB Dimension of $m \times n$ Matrices: Finding Basis
Replies
2
Views
2K
A Finding Eigenvectors for Two Matrices using the Generalized Jacobi Method
Replies
7
Views
2K
A The meaning of ##(ict, x_1,x_2,x_3)##
Replies
4
Views
2K
I Standard basis and other bases...
Replies
9
Views
3K

Hot Threads

Recent Insights

Back
Top