Math 311h - Test I Review

General Information

Test I (Wednesday, October 1) will have 5 to 7 questions, some with multiple parts. It will cover chapters 1 and 2, and sections 3.1-3.2A. In addition, it will cover the material contained in my Notes on Row Reduction. Please bring an 8½×11 bluebook. Problems will be similar to ones done for homework. You may use calculators to do arithmetic, although you will not need them. You may not use any calculator that has the capability of doing either calculus or linear algebra.

Topics Covered

Vectors

Geometry of Rⁿ
- linear combination of a set of vectors
- dot product
- Schwarz's inequality
- triangle inequality
- norm or length of a vector
- angle between vectors
2D and 3D geometry
- Be able to find the equations of lines and planes, especially their parametric equations. Be familiar with the properties of the dot and cross products in 2D and 3D. Be able to compute areas of paralleograms and volumes of parallelepipeds using cross and dot products.

Systems & matrices

Linear systems
- Solving systems via row reduction.
  - Augmented matrix form. Convert a system to and from augmented matrix form.
  - Row operations and equivalent systems. Be able to define the term equivalent system. Know the three types of row operations that give equivalent systems:
    1. elementary multiplication
    2. elementary modification
    3. elementary interchange
  - Row echelon form of a matrix and reduced row echelon form. Be able to use Gauss elimination to put a matrix in row echelon form. Be able to identify the lead variables and free (nonleading) variables. Be able to use Gauss-Jordan reduction to put a matrix in reduced row echelon form. (This form makes the connection between lead variables and free variables explicit.) Be able to find all solutions of a linear system by row reducing its augmented matrix and reading off the solution to the resulting equivalent system. See the Notes on Row Reduction for examples.
- Various types of systems: homogeneous, consistent, inconsistent, overdetermined, underdetermined. Be able to determine whether a system has a solution, and, if it does, how many solutions it has, by comparing the rank of A and [A|b].
Matrices
- Matrix algebra. Sum, product, scalar multiples, row vectors, column vectors, transpose, symmetric matrix, identity matrix, zero matrix, size of a matrix, (i,j) entry, notation. Know the "basic matrix trick"
  Ax = x₁a₁ + x₂ a₂ + ... +x_n an
  where the a_j's are the columns of A.
- Inverse of a matrix. Know how the inverse is defined. Also, know the terms invertible, nonsingular, and singular. Be able to be find the inverse of a matrix or show that a matrix is singular via row reducing [A|I]. Again, see the Notes on Row Reduction.
- Homogeneous systems. Know the connection with solutions to a general system and the corresponding homogeneous system.
  - Linearly independent and linearly dependent sets of vectors. Be able to test whether a set of vectors is LI or LD.
  - k-planes and solutions to linear systems. Be able to define the term k-plane and know how it relates to solutions of a linear system of equations.
Definition of row equivalence of matrices.
Be able to show these are quivalent conditions:
1. A is nonsingular.
2. Ax = 0 has only x = 0 as a solution.
3. A is row equivalent to I.
4. The columns of A are linearly independent.

Determinants

Basic properties. Know the basic properties for determinants. Be able to calculate the determinat of a matrix via its cofactor expansion about a row or a column.
- Determinants of special matrices. The determinant of an upper triangular, lower triangular, or diagonal matrix is the product of the diagonal entries. Also, det(A')=det(A).
- Row and column operations. Be able to use row operations to find a determinant.
- Row reduciton of a matrix A and det(A). Be able to read off the determinant of a matrix from the row operations used to reduce it and its row echelon form.
- Inverses. Be able to determine whether an n×n matrix A is invertible from knowing det A.
- Product rule. det(AB)=det(A)det(B).
Cramer's rule Be able to show that Cramer's rule is true. Be able to find the inverse of 2×2 and 3×3 matrices.

Linear functions on Rⁿ

Be able to define the terms linear function on Rⁿ, domain, image, range, one-to-one function, and inverse function.
Know and be able to state, prove, and use the representation theorem (pg. 106).
Know the connection between composion and matrix multiplication, function addition and matrix addition, scalar multiplication of functions and of matrices.
Be able to find matrix representations similar to ones we've done in class.

Vector spaces

Basic ideas. Addition, multiplication by scalars, and being closed under addition and scalar multiplication. Notation for special spaces: Rⁿ, R^m×n, P_n, C[a,b], C^k[a,b].