help with some general proofs and other questions involving vector spaces

• Feb 19th 2011, 01:19 PM
battleman13
help with some general proofs and other questions involving vector spaces
Determine wether the set, together with the indicated operations, is a vector space. If not identify atleast one of the ten vector space axioms that fail.

#1

The set of all 2x2 matrices of the form $\displaystyle \begin{array}{cc}a&b\\c&1\end{array}$ with the standard operations.

#2

The set of all 2x2 nonsingular (invertable or non zero determinant) with the standard operations.

I've wracked my brains with these for well over an hour now and made little to no progress... my book is of little to no help.

Thanks in advance for anyone that can help! (Hi)
• Feb 19th 2011, 01:40 PM
Plato
Quote:

Originally Posted by battleman13
Determine whether the set, together with the indicated operations, is a vector space. #1 The set of all 2x2 matrices of the form $\displaystyle \begin{array}{cc}a&b\\c&1\end{array}$ with the standard operations.

#2 The set of all 2x2 nonsingular (invertable or non zero determinant) with the standard operations.

Do either of those contain the zero vector?
• Feb 20th 2011, 11:35 AM
battleman13
As far as I understand it, both of those do contain the zero vectors.
• Feb 20th 2011, 11:55 AM
DrSteve
They do not!. The zero square matrix of order 2 has a zero for each entry - it does not have a 1 for the (2,2) entry.

For the second one, the zero matrix is not invertible.