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Show transcribed image text Problem 4. (40 points) For each of the following four statements, determine whether the statement is TRUE or FALSE. If the statement is true, give a proof. If the statement is false, give a counterexample. You will receive NO credit if you do not give a proof or a counterexample. (1) A matrix M is invertible if and only if its transpose MT is invertible. (2) Let M be an n × n diagonal matrix. Then the dimension of the space of solutions of a homogeneous linear system of equations with coefficient matrix M is equal to the number of diagonal entries of MM that are equal to zero. (3) Let N be a k × n matrix of rank r. Let M be an m × k ma- trix. The matrix MN has rank r if and only if NullSpace(M)n ColumnSpace(N) = {0} sequence is called a short exact sequence of finite dimensional vector spaces if Vi are finite dimensional vector spaces and hi are homomorphisms such that hi : Vi → ½ is one-to-one (injective), h2 : ½ → VS is onto (surjective) and the kernel of h2 equals the image of hı. For a short exact sequence of finite dimensional vector spaces, dim(V)-dim(½) + dim(V) = 0.