NAME:1 /15 2 /6 3 /16 4 /10 5 /10 6 /8 7 /10 T /75MATH 430 (Fall 2008) Exam 1, Sep 29No calculators, books or notes! Show all work and give complete explanations.This 75 minute exam is worth a total of 75 points.(1) [15 pts](a) Define the nullspace and range of a matrix.(b) State the Rank and Nullity Theorem, and illustrate what it says in the context of a well-chosen example.(c) Define the concept of a maximal linearly independent subset of a finite dimensional vector space.(2) [6 pts] Let A be a block matrix of the form A =BC. Prove that N (A) = N (B) ∩ N (C).2(3) [16 pts] Let A be the matrixA =1 2 34 9 73 7 48 18 14.Find bases for the four fundamental subspaces of A.3(4) [10 pts](a) Prove that if a matrix is both symmetric and skew-symmetric then it is zero.(b) Without using matrices prove that the composition of two linear mappings between vector spaces islinear.4(5) [10 pts](a) Let A be m×n and B be n×`. Prove that each column of AB can be expressed as a linear combinationof the columns of A. In particular, find the coefficients in these linear combinations.(b) LetA =1 23 45 6B =7 8 9 1011 12 13 14Use the formula you derived in (a) to calculate the 3rd column of AB.5(6) [8 pts] For each of the following statements either prove that the statement is true or give a specificcounterexample.(a) The union of two vector subspaces of a vector space is a vector subspace.(b) The intersection of two vector subspaces of a vector space is a vector subspace.6(7) [10 pts] Find a basis for the vector space consisting of all 3 × 3 skew-symmetric matrices and provethat it is indeed a basis.Pledge: I have neither given nor received aid on this