|Aug4-11, 12:41 AM||#1|
(LinearAlgebra) all 2x2 invertible matrices closed under addition?
1. The problem statement, all variables and given/known data
Suppose V is a vector space.
Is the set of all 2x2 invertible matrices closed under addition? If so, please prove it. If not, please
provide a counter-example.
2. Relevant equations
3. The attempt at a solution
well i know that what does it mean to be closed under addition. When V is closed under addition, if I suppose vector u and w are in the V, their addition u+w is also in the V, right?
The answer for the question is No.
A counter-example my professor provided is I+(-I)=0
I and (-I) are invertible, but their addition 0 is not invertible. and I know why it's not invertible.
But I don't figure out why it is not closed under addition,,.
If the addition is not invertible, does it mean that the addition is not in the V?
|Aug4-11, 02:22 AM||#2|
|Aug4-11, 03:53 AM||#3|
So clear, easy to understand. Thanks!
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