1 Linear Equations In Linear Algebra 2 Matrix Algebra 3 Determinants 4 Vector Spaces 5 Eigenvalues And Eigenvectors 6 Orthogonality And Least Squares 7 Symmetric Matrices And Quadratic Forms 8 The Geometry Of Vector Spaces 9 Optimization (online) 10 Finite-state Markov Chains (online) expand_more
1.1 Systems Of Linear Equations 1.2 Row Reduction And Echelon Forms 1.3 Vector Equations 1.4 The Matrix Equation Ax = B 1.5 Solution Sets Of Linear Systems 1.6 Applications Of Linear Systems 1.7 Linear Independence 1.8 Introduction To Linear Transformations 1.9 The Matrix Of A Linear Transformation 1.10 Linear Models In Business, Science, And Engineering Chapter Questions expand_more
Problem 1PP: Each of the following equations determines a plane in 3. Do the two planes intersect? If so,... Problem 2PP: Write the general solution of 10x1 3x2 2x3 = 7 in parametric vector form, and relate the solution... Problem 3PP: Prove the first pan of Theorem 6: Suppose that p is a solution of Ax = b, so that Ap = b. Let vh, be... Problem 1E: In Exercises 1-4, determine if the system has a nontrivial solution. Try to use as few row... Problem 2E: In Exercises 1-4, determine if the system has a nontrivial solution. Try to use as few row... Problem 3E: In Exercises 1-4, determine if the system has a nontrivial solution. Try to use as few row... Problem 4E: In Exercises 1-4, determine if the system has a nontrivial solution. Try to use as few row... Problem 5E: In Exercises 5 and 6, follow the method of Examples 1 and 2 to write the solution set of the given... Problem 6E: In Exercises 5 and 6, follow the method of Examples 1 and 2 to write the solution set of the given... Problem 7E: In Exercises 7-12, describe all solutions of Ax = 0 in parametric vector form, where A is row... Problem 8E: In Exercises 7-12, describe all solutions of Ax = 0 in parametric vector form, where A is row... Problem 9E: In Exercises 7-12, describe all solutions of Ax = 0 in parametric vector form, where A is row... Problem 10E: In Exercises 7-12, describe all solutions of Ax = 0 in parametric vector form, where A is row... Problem 11E: In Exercises 7-12, describe all solutions of Ax = 0 in parametric vector form, where A is row... Problem 12E: In Exercises 7-12, describe all solutions of Ax = 0 in parametric vector form, where A is row... Problem 13E: Suppose the solution set of a certain system of linear equations can be described as x1 = 5 + 4x3,... Problem 14E: Suppose the solution set of a certain system of linear equations can be described as x1 = 3x4, x2 =... Problem 15E: Follow the method of Example 3 to describe the solutions of the following system in parametric... Problem 16E: As in Exercise 15, describe the solutions of the following system in parametric vector form, and... Problem 17E: Describe and compare the solution sets of x1 + 9x2 4x3 and x1 + 9x2 4x3 = 2. Problem 18E: Describe and compare the solution sets of x1 3x2 + 5x3 = 0 and x1 3x2 + 5x3 = 4. Problem 19E: In Exercises 19 and 20, find the parametric equation of the line through a parallel to b. 19. a =... Problem 20E: In Exercises 19 and 20, find the parametric equation of the line through a parallel to b. 20. a =... Problem 21E: In Exercises 21 and 22, find a parametric equation of the line M through p and q. [Hint: M is... Problem 22E: In Exercises 21 and 22, find a parametric equation of the line M through p and q. [Hint: M is... Problem 23E: a. A homogeneous equation is always consistent. b. The equation Ax = 0 gives an explicit description... Problem 24E: a. If x is a nontrivial solution of Ax = 0, then every entry in x is nonzero. b. The equation x =... Problem 25E: Prove the second part of Theorem 6: Let w be any solution of Ax = b, and define vh = w p. Show that... Problem 26E: Suppose Ax = b has a solution. Explain why the solution is unique precisely when Ax = 0 has only the... Problem 27E: Suppose A is the 3 3 zero matrix (with all zero Describe the solution set of the equation Ax = 0. Problem 28E: If b 0, can the solution set of Ax = b be a plane through the origin? Explain. Problem 29E: In Exercises 29-32, (a) does the equation Ax = 0 have a nontrivial solution and (b) does the... Problem 30E: In Exercises 29-32, (a) does the equation Ax = 0 have a nontrivial solution and (b) does the... Problem 31E: In Exercises 29-32, (a) does the equation Ax = 0 have a nontrivial solution and (b) does the... Problem 32E: In Exercises 29-32, (a) does the equation Ax = 0 have a nontrivial solution and (b) does the... Problem 33E: Given A = [2672139], find one nontrivial solution of Ax = 0 by inspection. [Hint: Think of the... Problem 34E: Given A = [4681269], find one nontrivial solution of Ax = 0 by inspection. Problem 35E: Construct a 3 3 nonzero matrix A such that the vector [111] is a solution of Ax = 0. Problem 36E: Construct a 3 3 nonzero matrix A such that the vector [121] is a solution of Ax = 0. Problem 37E: Construct a 2 2 matrix A such that the solution set of the equation Ax = 0 is the line in 2 through... Problem 38E: Suppose A is a 3 3 matrix and y is a vector in 3 such that the equation Ax = y does not have a... Problem 39E: Let A be an m n matrix and let u be a vector in n that satisfies the equation Ax = 0. Show that for... Problem 40E: Let A be an m n matrix, and let u and v be vectors in n with the property that Au = 0 and Av = 0.... format_list_bulleted