Category Archives: linear algebra

Review exercise 2.6. Given the matrices find bases for each of their four fundamental subspaces. Answer: The second column of is equal to twice the first column so the rank of (and the dimension of the column space of ) … Continue reading →

Review exercise 2.5. Given the matrices find their ranks and nullspaces. Answer: We can use elimination to reduce to echelon form. We first exchange the first and third rows: and then subtract 1 times the second row from the third … Continue reading →

Review exercise 2.4. Given the matrix find its echelon form and the dimensions of the column space, nullspace, row space, and left nullspace of . Answer: We perform elimination on to reduce it to echelon form. In the first step … Continue reading →

Review exercise 2.3. State whether each of the following is true or false. If false, provide a counterexample. i) If a subspace is spanned by a set of vectors through then the dimension of is . ii) If and are … Continue reading →

Review exercise 2.2. Find a basis for a two-dimensional subspace of that does not contain , , or . Answer: One approach is to come up with a linear system that has a two-dimensional nullspace that excludes the coordinate vectors. … Continue reading →

Review exercise 2.1. For each of the following subspaces of find a suitable set of basis vectors: a) all vectors for which b) all vectors for which and c) all vectors consisting of linear combinations of the vectors , , … Continue reading →

Exercise 2.6.21. Consider the transformation from to that takes into . What is the axis of rotation for the transformation? What is the angle of rotation? Answer: We can approach this problem in at least two ways. The first way … Continue reading →

Exercise 2.6.20. A nonlinear transformation from a vector space to a vector space is invertible a) if for any in there exists some in such that and b) if and are in then implies that . Describe which of the … Continue reading →

Exercise 2.6.19. Let be the vector space consisting of all cubic polynomials of the form and let be the subset of consisting of only those cubic polynomials for which . Show that is a subspace of and find a set … Continue reading →

Exercise 2.6.18. Given a vector in find a matrix that produces a corresponding vector in in which all entries are shifted right one place. Find a second matrix that takes a  vector in and produces the vector in in which … Continue reading →