My Math Forum Proving matrices form a vector space

 Abstract Algebra Abstract Algebra Math Forum

 March 8th, 2012, 03:30 PM #1 Newbie   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
 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)).

 Tags form, matrices, proving, space, vector

,

,

# show the the matrices eij from a basis for m2

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post rain Linear Algebra 11 April 14th, 2013 02:59 PM baxy Advanced Statistics 0 March 14th, 2013 05:07 AM bjorno Real Analysis 0 March 10th, 2012 01:14 AM Sol Linear Algebra 4 September 17th, 2011 07:14 AM reddmann Linear Algebra 1 July 30th, 2007 09:47 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top