Does set of inv. 2-by-2 matrices constitute vector space?

[moy1989]

Hey guys, how could I get started on the following problem?

Does the following set of vectors constitute a vector space? Assume "standard" definitions of the operations.

The set of all invertible 2 X 2 matrices.

 

[tkhunny]

Re: Vector Spaces

List the properties necessary for a vector space.

Prove that the set you have indicated has those properties OR prove that it is missing at least one required property.

The very first thing to do is write the list of necessary properties.

 

[moy1989]

okay, I think I understand that one now.
I am still confused about how the set of all polynomials of degree 2 fail to constitute a vector space. The vectors fail the following conditions:

Vector spaces must be closed under addition
Vector spaces must be closed under scalar multiplication
There must be a zero vector

I understand how they fail the addition closure: ( x^2+x+1 ) + (-x^2+x+1) = 2x + 2, which is not a degree 2 polynomial.

There is not a zero vector of degree 2.

But what about scalar multiplication?
Would a counter example be if
c = 0
then, c( x^2 + x + 1 ) = 0, which is not a vector of the polynomials of degree 2?

 

[pka]

I fear that you have some grave missunderstandings.
The polynominal \(\displaystyle \left( {2x + 2} \right) \in P_2 \,\& \,\,0 \in P_2\).
For you see \(\displaystyle P_2\) is the set of all polynomials of degree less than or equal to 2.