# Midterm 2, Wednesday Nov. 14

I have scheduled some extra office hours for next week:
• Monday Nov. 12, 11-12 and 3-4.
• Tuesday 10-12 (usual hours) and 4-5
• Wednesday 11-12.

### Suggestions for studying:

The midterm will cover everything we've done through Wednesday's class , including 5.3 and 6.1. Instead of a list of problems as on the previous review sheet, here are some suggestions for studying using the book. You should review both the book and your notes--if I said something many times, it must mean that I think it's important! I would also review old HW problems as well as the first midterm and earlier quizzes.

Chapter 2: Make sure you really understand the relationship between a linear function and its matrix, and the relationship between rref(A) and issues such as when a linear transformation is one-to-one or onto. Review the relationship between composition and matrix multiplication, and the basic geometric linear transformations such as rotations, projections, reflections.

#### Some specific problems:

2.2/43,44 (for cross products)
2.3/31,33,35,40
2.4/22,23,25,28,46 (you can use stuff from 3.4 if you want),71
Review problems, pp. 94-5

Chapter 3: You should know the definitions of kernel and image, and be able to work with these concepts both computationally and abstractly. Likewise for subspaces, bases, linear independence, dimension, etc. You should be able to use the rank + nullity formula and other ideas connecting rref with dimensions of various subspaces like the kernel or image. Review coordinates with respect to the standard basis and other bases, and the matrix of a linear transformation with respect to an arbitrary basis. Good examples are geometric ones such as reflections, etc. Relation between matrices with respect to different bases; similarity of matrices.

#### Some specific problems:

3.1/30,31,33,34,35,39,51,52
3.2/24,34,36,50,51
3.3/13,15,26,27,39,42,43,45
3.4/7,9,15,18,20,44,46 (review 45),51
Review problems, pp. 146-147
Chapter 5: Review ON bases and their use in finding projections onto a subspace and orthogonal to a subspace, and reflections. Geometry via dot products: lengths and angles, Cauchy-Schwarz and triangle inequalities as well as Pythagorus' theorem. Using dot products to find coordinates with respect to an ON basis. Finding ON bases using Gram-Schmidt. Transpose and orthogonal matrices.

#### Some specific problems:

5.1/11,12,17,19,21,29,31,39
5.2/13,31
5.3/2 (using problem 1),5,7,8,13,17,19,24,25
Chapter 6: You should know how to compute 2x2 and 3x3 determinants, and the definition of determinants in terms of patterns as in 6.1. For the 2x2 case, the relationship between determinants and invertibility.

#### Some specific problems:

6.1/1-20 (or until you feel comfortable with the definition!),21,22,26,33,35,40