Quizzes 1, 2, 3

There will be a series of short quizzes on basic topics at the beginning of each class for a while.

· Wed 10/29:  Solving the Homogeneous Equation Ax = 0 and (in the new language) finding a basis for the null space of A.

You will be given a matrix A and asked to write the solutions of Ax = 0 in vector form (the review part).  You will also be asked to use your answer to write down a basis for the null space of A. (the new part, but if you have the vector form, you can write it right down).

What you can study:  Example 3, p. 192; Example 2, p. 49; Problem 45, p. 59 (answer in the back of the book), Examples 2 and 3, pp. 180-1.

· Fri 10/31: Linear independence

You will be given either a set of vectors (maybe a list of vectors, or maybe the columns of a matrix A or the rows of matrix B) and asked to determine whether or not they are linearly independent.

What you can study:  Examples 2 and 3, pp. 73-74; also the early problems in Exercises, Section 1.7.  Be sure to consider how to check independence of row vectors (same as columns of the transpose).

· Mon 11/3: Specification Equations for a Span

You will be given a set V of vectors in n-space (maybe a list of vectors, or maybe the columns of a matrix A or the rows of matrix B) and asked to find an algebraic specification of the span Sp(V).  This means finding a set of homogeneous linear equations whose solution set is W.  IMPORTANT: If the span of V is all of n-space, then there are no equations (or you could say the empty set of equations).  If the vectors in V are the columns of a matrix A, the span of V is (by definition) the Range of A, which is the set of vectors b for which Ax = b is consistent.

What you can study:  Example 1, pp. 178-9; Examples 1 and 2, pp. 190-1; some of Exercises 27-37 (odd), p. 187 (answers in the back of the book).