Math 415. Appled Linear Algebra
Midterm 3 Checklist

Chapter 6. Equilibrium

6.2. Electrical Networks

• Know how to set up and solve an electrical network problem in which an external current vector is present
• Know how to set up and solve an electrical network problem in which a battery vector is given
• How do you ground a node and what does this due to the solution method?

Chapter 7. Linearity

7.1. Linear Functions

• Know the definition of lineaity of a function and how to very if it in simple cases (review exercises 7.1.1, 7.1.2, 7.1.3 and 7.1.19)
• Theorem 7.5 is vital since it characterizes all linear functions from Rⁿ to R^m. Be able to find matrix representations for a variety of vector spaces.
• Know Lemma 7.11: The matix corresponding to a composition is the product of the separate matrix representation
• Know Definition 7.14 and Lemma 7.15: The matrix corresponding to the inverse of a linear function is the inverse of its own matrix representation. Be able to compute inverses this way.

7.2. Linear Transformations

• Be able to construct geometrically simple examples of rotations, reflections, stretches and shears in 2 dimentions
• Know the change of basis results characterized in Eqns (7.27) and (7.28) and how to use them in practice. See Handout 3 for an example

Chapter 8. Eigenvalues

8.2. Eigenvalues and Eigenvectors

• Know your definitions: eigenvalue, eigenvector, eigenspace, characteristic polynomial, multiplicity, trace, etc.
• Know how to find the eigenvalues and eigenvectors of a matrix in simple cases

8.3. Eigenvector Bases and Diagonalization

Diagonalization is one of the most important products of linear algebra. It is used everywhere in linear theory to "decouple" systems of equations. Know the leading results: a) eigenvalues corresponding to distinct eigenvalues are lin indep, b) if A is complete, then there is an invertible S and a diagonal D such that S^-1AS = D. How do you find S and D? This is what we refer to as "diagonalizing A".