My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum

LinkBack Thread Tools Display Modes
March 8th, 2012, 03:30 PM   #1
Joined: Jan 2012

Posts: 3
Thanks: 0

Proving matrices form a vector space

This problem asks us to show that matrices e i,j and Root(2)e i,j (1</= i, j</=2) form a basis for M2(Q[Root(2)]) considered as a Q-vector space.
It also asks to show the basis for M2(C) considered as a vector space over R.

I'm pretty lost when it comes to showing that something is a basis, the topic has not been covered well and the book doesn't do it justice. The only examples I can find online are concrete which makes it harder work out.

Thanks so much for the help.
lteece89 is offline  
March 13th, 2012, 09:50 PM   #2
Senior Member
Joined: Mar 2012

Posts: 294
Thanks: 88

Re: Proving matrices form a vector space

i assume you mean Eij is the matrix with 1 in the ij-th entry and 0's elsewhere.

to prove these 8 matrices form a basis for V = Mat(2,Q(?2)), we need to show 2 things:

they span V, and they are linearly independent.

suppose we have an arbitrary element A of Mat(2,Q(?2)): A =

[a b]
[c d].

since a,b,c,d are in Q(?2), we can write this as: A =

[s+t?2 u+v?2]
[w+x?2 y+z?2], where s,t,u,v,w,x,y and z are all rational.

thus A = sE11 + uE12 + wE21 + yE22 + t(?2E11) + v(?2E12) + x(?2E21) + z(?2E22), which shows the 8 matrices span V.

to prove linear independence, we need to show that if:

sE11 + uE12 + wE21 + yE22 + t(?2E11) + v(?2E12) + x(?2E21) + z(?2E22) is the 0-matrix, we MUST have:

s = t = u = v = w = x = y = z = 0.

start by "collecting terms", if such a linear combination is 0, it must be that:

(s+t?2)E11 + (u+v?2)E12 + (w+x?2)E21 + (y+z?2)E22 = 0 (*).

look at the 1,1 entry, the only matrix to contribute non-zero terms is E11. since the 1,1 entry of E11 is 1,

if the 1,1 entry of the sum (*) is 0, it must be the case that s+t?2 = 0. now show this means s = t = 0.

proceed similarly for the remaining 3 coordinates in the 2x2 matrix.

for M2(C), it might help to think of C as R(?(-1)).
Deveno is offline  

  My Math Forum > College Math Forum > Abstract Algebra

form, matrices, proving, space, vector

Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Need help with proving vector space over corpus rain Linear Algebra 11 April 14th, 2013 02:59 PM
Proving that mismatches increase matching space baxy Advanced Statistics 0 March 14th, 2013 05:07 AM
Proving that product K x K of a space K is compact bjorno Real Analysis 0 March 10th, 2012 01:14 AM
proving properties of Matrices Sol Linear Algebra 4 September 17th, 2011 07:14 AM
Normal form of Congruent Matrices? reddmann Linear Algebra 1 July 30th, 2007 09:47 AM

Copyright © 2018 My Math Forum. All rights reserved.