This sheet specifies a selection of exercises from the text, and in some cases a few additional exercises, for the first several weeks of class. The notation 1-33o means the odd problems between 1 and 33, inclusive.   Any * problems in an assignment should be kept separate from the problems without a *.  Please follow these format instructions .

Section 1.1   1,3,5,7,13,15,17,21,23,25,29,33*, plus the following.

101. Give correct definitions of the following terms or expressions: linear system, solution set, equivalent, inconsistent, row equivalent.
102. Give an example of a system of 2 linear equations in 2 unknowns with no solution. 
103. (*) Can a system of linear equations have just one equation?
104. (*) Is the following a linear system?

x2 + y2	=	13
x2 -y2	=	5

Explain why someone might say that the system is linear in x² and y².

105. Is it possible for a linear system of two equation in two unknowns to have exactly 1 solution? Exactly 2 solutions? Exactly 100 solutions? An infinite number of solutions? For each question, if you say it is not possible, explain why. If you say it is possible, give an example.

101. Two students are doing linear algebra homework. They each start with the same matrix and use row operations to obtain echelon form. When they compare their work, they see that they did not get the same answer. Did someone make an error?
102. One of the two students of the previous problem suggests, Let's keep going until we get to reduced echelon form. Then if one of us made an error we will know for sure because we will still have two different answers. Do you agree with this reasoning?
103. (*) Why do you think that the book defines pivot position and pivot column, but does not define pivot row?
104. How can you tell by looking at the reduced echelon form of the augmented matrix of a linear system whether or not there are any free variables? How can you tell which variables are the free variables and which are the basic variables?
105. (*) If a system has more variables than equations, is it possible for it to have a unique solution?

101. (*) Can  be expressed as a linear combination of  and  ?
102. Can  be expressed as a linear combination of  and  ?
103. Define linear combination. Explain what is meant by the set of vectors spanned by another set of vectors.
104. In the blue box on page 34, the vector equation
     
     

x1 a1 + x2a2 + ... + xnan = b

appears. In this equation, which letters are the variables? Which stand for constants? Give a specific example of such an equation with n = 4, and with the variables left as letters, but with the constants replaced with some specific values.

105. (*) Suppose that vectors u, v, and w are given, and you notice that w = u+ v. Explain why every vector in Span {u, v, w} is also in Span{u, v}.
106. (*) If it is true that every vector in Span {u, v, w} is also in Span{u, v}, must w be somehow made up from u and v, as in the last problem? Explain. (Note, do not assume that the vectors in this problem are the same ones described in the preceding problem.)

101. At the middle of page 43 there is an example in which an augmented matrix has a final column given by  . This final column can be written as a product Cb where C is a 3 by 3 matrix, and b is the vector  . Find the matrix C.
102. Proving properties of matrix operations: In general, to prove properties like those shown in theorem 5, you must prove that two things are equal, namely, the final result on either side of the equal sign. When the two items are vectors, the usual way to do this is to develop a formula for the entry in position j (called the j-th entry) of each side, and show that they are equal. Do this for the two equations in theorem 5, using the following fact: if the ij entry of the m × n matrix A is a_ij and if the j^th entry of vector v is v_j then for any i, the i^th entry of Av is given by .

Section 1.5   1,3,5-7,11-15,17-23o,25*, 27*, 29-33, 40*, plus the following:

101. Define homogeneous system, trivial solution, nontrivial solution, parametric vector equation
102. (*) Three students are independently working on solving an inhomogeneous linear system of the form A x = b. The first student finds a solution vector x. The second student finds a different solution vector y. Did someone make an error? A third student finds yet another solution, z. The three students notice that x+ y = z. Did someone make an error?
103. (*) Prove: If a and b are solutions to a homogeneous system, then so is a + b.
104. Prove: If a is a solution to a homogeneous system, then so is ra for any real number r.
105. (*) Prove: If a and b are solutions to a homogeneous system, then so is ra+ sb, for any real numbers r and s.