Math 415. Appled Linear Algebra
Midterm 1 Checklist

Chapter 1. Linear Algebraic Systems

1.1. Solution of Linear Systems and 1.2. Matrices and Vectors

• Know how to add, scalar multiply, multiply matrices
• What is the identity matrix
• Review equation (1.11) and know why it is true

1.3. Gaussian Elimination - Regular Case

• Be able to perform Gaussian Elimination on both regular and augmented matrices
• Know how to solve systems using back substitution from Gaussian reduced form
• What is a regular square matrix?
• What is an elementary matrix of the first type, i.e. how do you form one and then what does it do?
• How do you get the inverse of an elementary matrix?
• What is meant by the LU factorization of a regular matrix?
• Be able to find the factorization efficiently for square regular matrices.
• Be able to solve Ax = b using back and forward substitution methods

1.4. Pivoting and Permutations

• What is the definition of "non-singular" and what is important about "non-singular" matrices?
• What is a permutation matrix? How do you form them and what do they do?
• Be able to find the PLU factorization of a square matrix and then solve Ax = b by foward and backward substitution

1.5. Matrix Inverses

• What is the definition of an "inverse"?
• What is the formula for the inverse of a 2 x 2? What is the determinant of a 2 x 2?
• What is the inverse of an inverse?
• What is the product rule for inverses?
• What is the PLDV factorization? Can you find it in simple cases?

1.6. Transposes and Symmetric Matrices

• What is the transpose of a matrix? What is the product rule for transposes?
• If A is invertible, is its transpose invertible?
• What are symmetric matrices and how does the LDV factorization change for them?

1.8. General Linear Systems

• Know what is meant by "row echelon form" and how to find it using Gaussian elimination
• Know how to find the general PLV factorization of a matrix
• What are pivots, basic variables and free variables?
• Know the definition of "rank" and how it is connected with "non-singulaity"
• How many solutions can a linear system have? Explain. What is the geometric version of your answer?
• How many solutions can a "homogeneous" system have?

1.9. Determinants

• Know the definition of determinant in terms of its properties
• Be able to compute determinants in simple cases
• How are "invertible", "non-singular", and "full rank" connected to values of the determinant?
• What are the product, inverse and transpose rules for determinants?

 

Chapter 2. Vector Spaces and Bases

2.1. Real Vector Spaces

• What are Rⁿ, P⁽ⁿ⁾, M_{m x n} and F(I) ? In particular, how are addition and scalar multiplication defined on these spaces?

2.2. Subspaces

• What is a subspace?
• How do you show that a subset of a vector space is a subspace?
• Know the geometric representation of subspaces given at the bottom of page 85

2.3. Span and Linear Independence

• Know what is meant by linear combination and span of a set of vectors
• Know what is meant by linear dependence and independence and how to verify one or the other in Rⁿ, M_{m x n} and P⁽ⁿ⁾
• Know Theorem 2.21
• Know and understand the statements of Lemma 2.23, Proposition 2.24 and Proposition 2.25

2.4. Bases and Dimension

• What is a basis of a vector space?
• Do all bases have the same number of vectors? What is this number called?
• What are the "coordinates" of a vector relative to a basis?
• What are the dimensions of Rⁿ and P⁽ⁿ⁾and why?
• Know how to find a basis for the subspace of solutions of one or more linear homogeneous equation.

2.5. The Fundamental Matrix Subspaces

• Know the definitions of adjoint system, kernel, cokernal, range, corange
• Be able to find bases for any of the four subspaces above
• There are two principal superposition results in this section, Propositions 2.37 and 2.39. Know what they say and how to use them and be able to replicate their proofs

2.6. Graphs and Incident Matrices

• Be able to turn a digraph into an incident matrix and an incident matrix into a digraph
• Know how to find a basis of independent circuits for a digraph
(review solution of problem 2.6.6)
• Know how to write other circuits in terms of your independent ones (review solution of problem 2.6.6)
• What is Euler's formula?