Vector Space

Jan 2010
79
1
Does the following set of vectors constitute a vector space? Assume "standard" definitions of the operations.

The set of all \(\displaystyle 2x2\) matrices with determinant equal to zero.

If not, which condition(s) below does it fail? (Check all that apply)

A. Vector spaces must be closed under addition
B. Vector spaces must be closed under scalar multiplication
C. There must be a zero vector
D. Every vector must have an additive inverse
E. Addition must be associative
F. Addition must be commutative
G. Scalar multiplication by 1 is the identity operation
H. The distributive property
I. Scalar multiplication must be associative
J. None of the above, it is a vector space

I know it's not a vector space, but I'm not sure which conditions it fails. Any insight is appreciated.
 
Oct 2009
4,261
1,836
Does the following set of vectors constitute a vector space? Assume "standard" definitions of the operations.

The set of all \(\displaystyle 2x2\) matrices with determinant equal to zero.

If not, which condition(s) below does it fail? (Check all that apply)

A. Vector spaces must be closed under addition
B. Vector spaces must be closed under scalar multiplication
C. There must be a zero vector
D. Every vector must have an additive inverse
E. Addition must be associative
F. Addition must be commutative
G. Scalar multiplication by 1 is the identity operation
H. The distributive property
I. Scalar multiplication must be associative
J. None of the above, it is a vector space

I know it's not a vector space, but I'm not sure which conditions it fails. Any insight is appreciated.

Check the sum of the matrices \(\displaystyle A=\begin{pmatrix}1&0\\0&0\end{pmatrix}\,,\,B=\begin{pmatrix}0&0\\0&1\end{pmatrix}\) , both of which belong to the set you define above...(Wink)

Tonio