## Math 272: Linear Algebra with Applications Study Guide for Exam 2

The second midterm exam is on Monday, April 10, during our regular class time.

The following is a chapter by chapter guide intended to help you organize the material we have covered in class as you study for your exam. Although this exam will emphasize chapters covered since the first midterm, it is still very important to know the earlier material as later chapters build on this material. The information here is only intended to serve as a guideline, and may not explicitly mention everything that you need to study.

2.2 & 2.3: You must know the definition of a linear combination, and be able to determine (and prove) whether a vector is a linear combination of a set of vectors or not. You must also know the definitions of linear independence and linear dependence, and how to prove whether a set of vectors is linearly independent or not. Finally, you should know the theorems discussed in class that associate these concepts with systems of equations, pivot positions in matrices and the invertibility of matrices.

3.1 & 3.2: You should know what we mean by an abstract vector space, which is defined by the 10 axioms given on p. 129. You should also know what we mean by a subspace, and how to check that a subset of a given vector is a subspace by only verifying axioms 1 and 6, or by using Theorem 4. You may also use the fact that a subset does not contain the zero vector to quickly determine that it is not a subspace. Know how to show that the span of a set of vectors is a vector space, and that the null space of a matrix is a vector space.

3.3: You should know that a basis is a linearly independent set that spans a space. You should also know how to determine whether a given set is a basis of a subspace, and how to find a basis of a subspace if none is given. Know how to use the basis to determine the dimension of the subspace. You should also know how to determine the rank and nullity of a matrix, and find its row space, column space and null space by finding a basis for each of these. Know the statement of the Rank-Nullity Theorem for matrices.

3.4 Given an ordered basis of a general vector space, you should be able to find the coordinate vector for a given vector in that space. You must also know how to find the transition matrix between two bases and how to use this to find the coordinate vectors in different bases.

4.1 & 4.4: Know what we mean by the terms transformation, domain, matrix transformation and linear transformation. Know how to determine whether a transformation is linear or not, and find the standard matrix of a linear transformation. You should also be able to determine whether a linear transformation is orthogonal or not, using the definition from our Mathematica lab. Be familiar with the standard examples of matrix transformations such as stretches by a given factor, reflections, and rotations (including rotation by a general angle θ). This test will not include problems using homogeneous coordinates.

4.2: Know what we mean by the kernel (or null space) and image (or range) of a transformation, and how to find them. Make sure you know the statements of the theorems in this chapter, including the Rank-Nullity Theorem applied to transformations.

4.3: Know what it means for a transformation to be one-to-one and onto and how to prove these properties in general, but especially for linear transformations. You should also know that an isomorphism of vector spaces is a transformation that is one-to-one, onto and linear.

Here is a handout made by Shelly Tang to explain linear independence, spanning sets and bases.

Please be sure to review homework and example problems for the chapters given above!

Below is a list of relevant additional practice problems for each section.

Section Suggested Problems
3.2 1-30, 37-50, 52
3.3 5, 7, 11, 12, 25-30, 37, 39, 43
3.4 3, 7, 11, 13, 17, 23
CT3 2-35
4.1 7-16, 22-26, 31-37
4.4 1-12
4.2 17-30, 35-42
4.3 1-10, 25-31
CT4 1-12, 22-29