# Math 2051 Midterm Review Packages

This 53 page Midterm 1 preparation guide includes examples and all theorems, lemmas, and corollaries dealing with the following topics:

Chapter 1 Material

1. Linear Combinations
2. Linear Independence and Linear Dependence
3. The Null Space of a Matrix
4. The Column Space of a Matrix
5. Chapter Recap

Chapter 2 Material

1. Vector Spaces and Subspaces
2. Linear Independence and Spanning Sets
3. Chapter Recap

Chapter 3 Material

1. Basis and Dimension
2. More on Rank and Nullity
3. Chapter Recap

Chapter 4 Material

1. One-Sides Inverses
2. Chapter Recap
3. All problems come with full solutions at a cost of \$35.00 + tax.

This 57 page Midterm 2 preparation guide includes examples and all theorems, lemmas, and corollaries dealing with the following topics:

Chapter 1 Material

1. Linear Transformations of Euclidean n-Space
2. Left Multiplication by a Matrix
3. Kernel and Range
4. One-to-One and Onto
5. Composition of Linear Transformations

Chapter 2 Material

1. The Matrices of a Linear Transformation
2. The Matrix of T Relative to V and W
3. Matrix Multiplication is Composition of Linear Transformations
4. Matrices of the Identity
5. Changing Coordinates
6. Similarity of Matrices

All problems come with full solutions at a cost of \$35.00 + tax.