Math 0 Test I M. Randall Holmes September 8, 008 This exam will begin at 9:40 am and end at 0:5 am. You may use your writing instrument, a calculator, and your test paper; books, notes and neighbors to remain firmly closed. If you solve an equation or system of equations with your calculator, I expect you to show what you did (and I expect you to make appropriate use of reduced echelon form).
. Solve the system of equations by converting it to matrix form and then using the algorithm studied in class to convert to reduced echelon form. Show each matrix and state the elementary row operations used to get each matrix. Do not use back substitution: read your final solution from the reduced echelon form. State the general solution to the problem (involving free variables) then state a specific solution to the problem (specific numbers that work). x + x x + x 4 = x + 5x 7x + 8x 4 = 5 x + 8x x + 4x 4 =
. Write down the augmented matrix representing the given system of equations. Write down the vector equation and matrix equation equivalent to the given system of equations. You do not need to solve the system. x + x = 4 x x + 7x = 0 x x = 5x x = 0
. In each part, a matrix is given in echelon form. Answer the following questions (and carry out operations if necessary) in each part. What are the basic variables and free variables in the system of linear equations with this augmented matrix? Does the system of linear equations with this augmented matrix have no solutions, a unique solution, or infinitely many solutions? Explain how you can tell without doing any calculations. Is the matrix in reduced echelon form? If it is, write the solutions to the corresponding system of equations in the usual way (a value for each variable, possibly depending on free variables) and then write out the solution set in parametric vector form. If it is not, carry out the elementary row operations required to reduce it to row echelon form, showing all matrices and describing all elementary row operations used. Then give the solution set of the corresponding system of equations in the usual form (a value for each variable, possibly depending on free variables) and in parametric vector form. (a) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4
(b) 0 0 0 0 0 0 0 5
4. A matrix in reduced echelon form is given. Give two matrices such that the corresponding systems of linear equations have the same solution set as the system corresponding to this matrix: the first matrix should be in echelon form but not in reduced echelon form, and the second should not be in echelon form. Label the two matrices echelon and not echelon. Hint: row operations do not change the solution set. 0 4 0 0 0 0 0 0 4 6
5. Is the given vector in the span of the given set of vectors? Write down and solve a vector equation for the weights in the linear combination of the elements of the given set which will evaluate to the given vector, and end by writing the given vector as a linear combination of the elements of the given set or explaining why this is not possible. The given vector: 9 5 The given set of vectors: 0,, 6, 4 7
6. A number of sets of vectors are given. For each set, determine whether it is linearly dependent or linearly independent. In some cases you should be able to determine linear dependence or independence very quickly; in such cases give a brief explanation of your answer. In each part, write the zero vector as a nontrivial linear combination of the vectors in the set if this is possible; in some easy cases you can just write it down, but if you need to solve a system of equations to find this combination (or to establish that there is no such combination) write down the matrix and its reduced echelon form. In each part, state the dimension of the span of the set of vectors (point, line, plane, -dimensional, etc.) (a) 4, 4 6 8 4 8
(b) 4, 0, (c),, 0 0 0 9
7. Set up and solve a system of linear equations representing the pictured traffic flow problem. The number of vehicles entering any intersection must be the same as the number of vehicles leaving it. Give a completely general solution (with free variables) then give the specific solution with x = 00. 0
8. Prove that if the matrix equation Ax = 0 has a nontrivial solution h 0, it is impossible for Ax = b to have a unique solution for any vector b (it might have no solutions or it might have infinitely many, depending on what b is, but it can t just have one) Hint: suppose that Ax = b has a solution x 0 0. Describe a second solution of Ax = b (and show calculations verifying that it is a solution).