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.

answer: the sun

explanation:

water cycle describes the evaporation of water from earth's surface, formation of clouds and precipitation water back onto the surface of earth. seventy percent of the earth is covered with water in the form of seas and oceans. the heat of the sun causes the evaporation of water from the surface of the ocean. the water vapors rise and condense to form water droplets. the collection of these water droplets form clouds which rain down again.

the evaporation of the oceans surface water as part of the water cycle is caused by the sun