Reading Response Questions:

Read the indicated section in preparation for class. Then, please draft a response to the Question(s) in full sentences. Email your response to emmons@pacificu.edu with the subject line

Math 306 RR for <due date>

Due by 11:00am sharp on the date indicated.

Due Date	Section	Question(s)
Fri Feb 2	1.2	1. Why might the reduced echelon form of a matrix be preferable to an (unreduced) echelon form? 2. Is it possible for a 3x2 matrix to have 3 pivot columns? What about a 2x3 matrix?
Mon Feb 5	1.3	1. Let u be a vector in the plane R^2. Could the “Parallelogram Rule for Addition” be potentially confusing when used to draw the vector u+u? 2. If u is a vector in R^3, explain what Span{u} looks like geometrically.
Wed Feb 7	1.4	1. Rewrite Theorem 4 to the specific case of A being a ‘row vector,’ i.e., a 1xn matrix. What is an easy way of restating condition d. of the Theorem in this case? 2. In linear algebra we will often say “I is the identity matrix” instead of “I is an identity matrix.” Why is this technically incorrect? On the other hand, why does this not usually cause trouble?
Mon Feb 12	1.5	1. What is the trivial solution to Ax = 0? Why is it called trivial? 2. If Ax = 0 has infinitely many solutions, what can we say about the number of solutions to Ax = b for other vectors b?
Wed Feb 14	1.7	1. On page 67 the characterizations of when a set of one or two vectors is linearly dependent or linearly independent are given. What about {}, a set of no vectors whatsoever? Would it make more sense to call this linearly independent or linearly dependent? Justify your answer. 2. Give an intuitive explanation of Theorem 8.
Mon Feb 19	1.8	1. Describe how an mxn matrix defines a function from R^n to R^m. 2. If we apply the linear transformation T from Example 5 to the sheep from Example 3, what would the resulting image look like? 3. Can the definite integral and/or the derivative be thought of as linear transformations (acting not on vectors in R^n, but on functions)?
Wed Feb 21	1.9	1. Suppose T is a linear transformation from R^3 to R^3. Explain how knowing the value of T on just 3 vectors is enough information to know T(v) for any vector v. 2. Can you think of a rigid (i.e. shape-preserving) transformation of the unit square in R^2 which cannot be captured by a 2x2 matrix, or is there nonesuch?
Fri Feb 23	2.1	1. Explain why matrix multiplication is not commutative (AB is not equal to BA) from the point of view of the fact that “Multiplication of matrices corresponds to composition of linear transformations” (pg 110). 2. If one wanted to consistently work with row vectors instead of column vectors, how could this be achieved with the transpose operator? What would the equation Ax=b look like after this transformation?
	2.2	1. If someone hands you a square matrix A, how would you go about finding its inverse? 2. Why is the notation A^-1b used as opposed to b/A?
	2.3	1. Suppose A is a square matrix and there exist matrices C and D satisfying AD = I and CA = I. Must is be that C = D? 2. Write down several conditions that are equivalent to A not being invertible.
Mon Mar 12	3.1 – 3.3	1. Make an analogy between computing the determinant of an 8x8 matrix via cofactor expansion and repeatedly placing 8 rooks on chessboard. What does the square being light/dark correspond to?
Wed Mar 14	4.1	1. Give several examples of vector spaces. Try to think of an example that isn’t in the book. 2. Is it accurate to say R^1 is a subspace of R^2? Discuss.
Wed Mar 21	4.2	1. Suppose A is an mxn matrix. Null(A) and Col(A) are both subspaces, but of what vector space(s)? 2. Can you think of a linear transformation from P2 to R^2, or is there none such?
Wed Apr 4	4.5	1. Name a 3 dimensional vector space which is not R^3. 2. Name an infinite dimensional vector space V. What specifically does it mean to say V is infinite dimensional? 3. Fix the following statement so it becomes true: Given a set of vectors S = {v1, …, vk} in R^n with k < n, this set may be uniquely expanded (by adding vectors) to a set which is a basis. [Hint: there are two errors with the statement.]
Wed Apr 11	5.2	1. Suppose A is an nxn matrix. Why is A invertible equivalent to the number 0 not being an eigenvalues of A? 2. True or false: The only matrix similar to the identity matrix is the identity matrix. (Justify your answer.)
Mon Apr 23	5.4	1. Explain how similar matrices can be thought of as representing the same linear transformation viewed under different bases. 2. Thinking in terms of the previous question, does it make sense that if A is diagonalizable the entries of the diagonal matrix are the eigenvalues?
Wed May 2	6.3	1. Let W be a two-dimensional subspace of R^3, fix a y not in W, and x be the vector y-hat (since I can’t typeset a y-hat here) from the Orthogonal Decomposition Theorem. What would be the radius of a sphere around y that just touches the plane W?