A)let x be the set of all 3x3 matrices whose diagonal entries are all 0. with the usual matrix addition and scalar multiplication, is x a vector space?

(b) let y be the set of all 3 x 3 matrices whose diagonal entries add up to 0. with the usual matrix addition and scalar multiplication, is y a vector space?

(c) let z be the set of all polynomials of degree exactly 5. is z a vector space?

d) let v be the set of all 3x3 matrices whose determinant is equal to 0. with the usual matrix addition and scalar multiplication, is v a vector space?

(e) let w be the vector space of all lower triangular 4x4 matrices. what is the dimension of w?

Identify the number 127 as a rational or irrational. explain

The input to the function is x and the output is y. write the function such that x can be a vector (use element-by-element operations). a) use the function to calculate y(-1.5) and y(5). b) use the function to make a plot of the function y(x) for -2 ≤ x ≤ 6.

Exclude leap years from the following calculations. (a) compute the probability that a randomly selected person does not have a birthday on october 4. (type an integer or a decimal rounded to three decimal places as needed.) (b) compute the probability that a randomly selected person does not have a birthday on the 1st day of a month. (type an integer or a decimal rounded to three decimal places as needed.) (c) compute the probability that a randomly selected person does not have a birthday on the 30th day of a month. (type an integer or a decimal rounded to three decimal places as needed.) (d) compute the probability that a randomly selected person was not born in january. (type an integer or a decimal rounded to three decimal places as needed.)