Question

Let V={[ a b

c 1 ]∣a,b,c∈R}
a. If addition and scalar multiplication are the standard componentwise operations, show that V is not a vector space.

Answer 1

To show that V is not a vector space, we can demonstrate that it does not have a zero vector, which is a required property for vector spaces.

Step-by-step explanation:

In order to show that V is not a vector space, we need to find at least one criterion of vector spaces that V fails to meet. One such criterion is the existence of a zero vector. The zero vector in V should satisfy the property that when added to any vector in V, the result is that same vector. However, in V, if we add the zero vector [0 0 0 1] to any other vector, the resulting vector will have a non-zero fourth component (1+1=2). Therefore, V does not have a zero vector, and it is not a vector space.