M314 Practice Exam 1

Note: These are mostly questions I've given on exams when I've taught M314 before, so this should give you an idea what my exams are like. And although it should help you to review, this is not a review sheet in the usual sense. There may be things you need to know that aren't on it, so be sure to review the book, the homework and the quizzes. On the other hand, problem 5 will be on the exam Wednesday (verbatim!), so be sure you know how to do problem 5.

Problem 1 (18 points): For each matrix A below, find its reduced row echelon form RA and determine its rank. Also, in each case, determine if the rows of the given matrix span Rn, where n in each case is the number of columns of the matrix, and determine if the rows of the given matrix are linearly independent. Show your work and explain your answers.
```(a) A=  [1   4   0   4  -1]
[0   0   1   4  -1]
[1   4   1   8  -2]

(b) A= [1   1    1]
[0   1   -1]
[0   1   -1]

(c) A= [1   2   3]
[1   0   4]
[0   2   0]
```

Problem 2 (15 points): Suppose
```[1  -6  0  0  3  | -2]
[0   0  1  0  4  |  7]
[0   0  0  1  5  |  8]
[0   0  0  0  0  |  0]
```
is the reduced echelon form of the augmented matrix of a system of 4 equations in the variables x1,..., x5.
• (a) Which variables are the free variables?
• (b) How many solutions does the system of equations have?
• (c) Write down the general solution of the system of equations in vector form.
Problem 3 (12 points): For each of the following statements, determine whether it is True or False; if it is false, give a specific example which shows it is not true (it is not enough to just say in general terms why it is false).
• (a) If A is the coefficient matrix of a linear system with infinitely many solutions, then the number of variables is more than the rank of A.
• (b) A linear system with fewer variables than equations never has infinitely many solutions.
• (c) A homogeneous linear system with a unique solution must have at least as many equations as variables.
• (d) A linear system with fewer equations than variables is never inconsistent.
• (e) Suppose a matrix has n columns, and that the matrix is taller than it is wide. Then the rows span Rn.
• (f) Suppose a matrix is taller than it is wide. Then the rows of the matrix are linearly dependent.
Problem 4 (15 points): Problem 21 p. 100. (On the actual exam, I will type out every problem.)
Problem 5 (15 points) Do Problem 58 on p. 27. Use complete sentences and good grammar in your answer!
Problem 6 (10 points) In Problem 14 on p. 14, write the vector from C to E as a linear combination of a and b.
Problem 7 (15 points) Consider the planes defined by x+y+z=2 and x+2y+z=0.
• (a) Find the cosine of the acute angle between the planes defined by x+y+z=2 and x+2y+z=0.
• (b) Find an equation (in vector form) for the line L where the two planes intersect.
• (c) Find (in normal form) an equation for the plane passing through the point (7,3,2) and perpendicular to L.