ubspace of V 8. Determine whether the following are linear transformation in P, the (a) L(p(x)) = x +p(x) for p e P. (b) L(p(x)) x2p(x) p(x) for peP. 9. (a) Show that L(f(x)) = f(x) + f(0) is a linear operator in C[-1,1 (b) Find kerL above. Find the range of L above. 10. Let S {(x1, 2, 3 , X4)| 31 + x2 = 3 + x4} be a subspace of R4. 11. Given v (1,-1, 1, 1) and w (4,2,2,1). (a) Determine the angle between v and w. b Find the orthogonal complement of V = span {v, w}. 12. Let A be an m x n matrix. (a) Suppose that rank A = r, what are dimensions of N(A) and b Verify that N(AT A) N(A) and rank(AT A) r.

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Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.