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nrev2 - Review Problems for Exam Two 1 Find the dimension...

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Review Problems for Exam Two 1. Find the dimension and a basis for the set S of symmetric 2 × 2 matrices. 2. Find the dimension and a basis for the set S of polynomials in P 4 p ( x ) such that p (2) = 0. 3. Suppose that the m × n matrices A and B are row equivalent. Which of the following are True/False: (a) A and B have the same rank (b) A and B have the same rowspace (c) A and B have the same column space (d) A and B have the same nullspace 4. Suppose the vector space V has dimension n and r n m . Which of the following are True/False: (a) Any set of m vectors must span V . (b) Any set of m vectors must be dependent. (c) Any set of r vectors must be independent. 5. Suppose that a 3 × 5 matrix A has rank 3. Which of (a) -(d) are True/False: (a) The rows of A linearly independent (b) The rows of A span R 5 (c) The columns of A linearly independent (d) The columns of A span R 3 (e) Find the dimension of the nullspace of A. 6. Let A be an m × n matrix, so that T A : F n F m . Which of the following statements is equivalent to the statement that T A is onto? (a) rk(A) = m (b) rk(A) = n (c) the columns of A span F m (d) the columns of A are independent (e) for each -→ b F m , there is at most one solution -→ x such that A -→ x = -→ b (f) for each -→ b F m , there is at least one solution -→ x such that A -→
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