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Practice for Exam 2 - Math 3130 Intro to Linear Algebra Spring 2016 - Dr. Radu Cascaval Exercise 1 Given the matrix 1 x x2 A = 1 2 4 , 1 3 9 (1) Compute det(A) by the co-factor expansion method. (2) Find all values of x such that A is not invertible. (3) Use elementary row operations to reduce A to a row echelon form. Recompute det(A) via this method. Exercise 2 Given the matrix 2 0 3 A = 0 3 2 , −2 0 −4 decide whether A is invertible or not. If it is, compute A−1 . Exercise 3 Solve the following linear system using Cramer’s Rule: x+y+z = 0 x − 2y + 2z = 4 x + 2y − z = 2 Exercise 4 (a) Find the complete solution of the linear system Ax = b, where 1 0 1 2 2 1 1 1 A= 3 0 1 , and b = 0 0 −3 3 9 (b) What is the rank of the matrix A? What is the column space of A? Exercise 5 Consider the matrix below (which happens to be the transpose of the matrix A in exercise 2(a).) 1 2 3 0 B = 0 1 0 −3 1 1 1 3 (a) Find rref(B). Is it related to rref(A) computed in exercise 2(a)? (b) What is the rank of B? Write down a basis for the column space of B. (c) Express the free column(s) of B as a linear combination of the columns listed in the basis found in (b). Exercise 6 Find a basis for the subspace of R4 consisting of all vectors that are orthogonal to the following three vectors v1 = [1, 1, 2, 0], v2 = [1, 1, 0, −1]T and v3 = [0, 0, 2, 1]T . Exercise 7 Determine whether the following four vectors span R3 or not. v~1 = [1, 4, 7]T , v~2 = [2, 5, 8]T , v~3 = [3, 6, 9]T , 1 v~4 = [1, 1, 1]T Exercise 8 Find a basis for each of the following subspaces of R4 . State the dimension of each subspace. (1) all vectors of the form (a, a + b, 0, 0), (2) all vectors of the form (a, b, b, b), (3) all vectors of the form (a, a, b, c) where a + b + c = 0. Exercise 9 Let (1) (2) (3) (4) (5) What What What What What 1 2 3 1 2 A = 2 4 6 2 4 . 3 6 9 1 2 is the rank of A? is a basis for the column space of A? are the dimensions of the row space and null space of A? is a basis for the row space of A? is a basis for the null space of A? Exercise 10 Which of the following subsets W of P2 (polynomials of degree 2 or less) are subspaces of P2 ? Why or why not? (1) all polynomials p(x) = ax2 + bx + c where a + b = 0 and c is any real number. (2) all polynomials p(x) = ax2 + c2 where a and c are any real numbers. Exercise 11 Review all TRUE/FALSE questions at the end of each section 4.1-4.5, 4.7, 4.8. Below are a few sample questions for exam 2: Determine whether each of the following statements are TRUE or FALSE. Justify your answer for full credit. (1) The set of invertible 2 × 2 matrices forms a vector space with usual operations of matrix addition and scalar multiplication. (2) If S is any subset of a vector space V , then span(S) is a subspace of V . (3) If S is a linearly independent set, then S does not contain the zero vector. (4) If V = span{v1 , . . . , vn } then {v1 , . . . , vn } is a basis for V . (5) If A and B are two n × n matrices that have the same row space, then A and B have the same column space. (6) The system Ax = b is consistent if b is in the column space of A. (7) Adding one additional column to a given matrix increases its rank by one.