Determine whether each set equipped with the given operations is a vector space. For those that are not vector spaces identify the vector space axioms that fail

1. The set of all 2 X 2 invertible matrices with the standard matrix addition and scalar multiplication.

2. The set of all real-line functions f defined everywhere on the real line and such that f (1) = 0 with the addition and multiplication operations

1) It is not a vector space

2) It is a vector space

Step-by-step explanation:

1) This is not a vector space. There is no neuter element for the addition, since the null matrix is not invertible.

2) This is a vector space. Lets denote the set given by this item A. A is, In fact, this is a subspace of the vector space given by the real-line funcitons defined in all the real line. Note that if k is a real number and f is an element of A, then

k*A is an element of A, because k*f is defined everywere (kf (x) = k * (f (x)))

and k*f (1) = k*0 = 0.

If f, g are elements of A, then f+g is an element of A: f+g (x) = f (x) + g (x) is defined everywhere and f+g (1) = f (1) + g (1) = 0+0 = 0.

Also, the zero function is definced everywhere and in 1 it takes the value 0. Since A is a subspace of the vector space given by the real line functions, then it s indeed a vector space.