Trying to Prove three 2x2 Matrices are Linearly Independent
Ive got a question asking;
The set of M2,2 of all 2x2 matrices, with real entries, is a vector space. For what values of A (real) is the set
a linearly independent subset of M2,2?
I have been trying to prove using a1V1+a2v2+a3v3=0 but I cant figure how to prove it or put it into a singular matrix form for row reduction... any help would be greatly appreciated.
Re: Trying to Prove three 2x2 Matrices are Linearly Independent
So you have
so we must have , , and only if .
Okay, have you tried solving for , , and ? If you subtract twice the first equation from the second, you eliminate - . If you subtract the third equation from the first you again eliminate - . Now eliminate by subtracting twice the first of those two equations from the other: .
If it follows that and . What is the only value of A such that does not have to be 0?