Math 423 - Test I Review

General Information

Test I (Tuesday, February 19) will have 5 to 7 questions, some with multiple parts. Paper will be provided. 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

Oscillatory systems and Least Squares - § 2.5, 2.6.
1. Be able to set up and solve simple spring mass systems.
2. Be able to find the best straight line fit to given data using least squares.
Solving linear systems and row reduction - § 3.1-3.4
1. Be able to row reduce a matrix. Know what leading entries, rows, and columns are.
2. Be able to solve linear systems via row reduction.
Elementary row operations and elementary matrices - § 3.5, 3.7
1. Know the connection between elementary row operations and the elementary matrices corresponding to them. Be able to show these matrices are invertible.
2. Be able to do an LU factorization, keeping track of elementary matrices involved.
Vector spaces and subspaces - § 5.1, 5.2
1. Know the definition of a subspace, and the conditions for a subset to be a subspace.
2. Know the basic examples of vector spaces we discussed - R^p, C^p, Pⁿ (polynomials of degree less than n), and C[a,b], the continuous functions on [a,b].
Linear independence, linear dependence, spans - § 5.3
1. Know definitions of these. Be able to check sets of vectors to determine whether they are LI or LD, or whether they span a given space.
Bases, dimension, coordinates and coordinate maps - § 5.4
1. Know the definitions of the concepts involved.
2. Be able to find the coordinates of a vector relative to an ordered basis.
3. Be able to check whether a set is a basis, given the dimension of a space and other information. For example, if the dimension of a space is 5, and the set has 7 vectors, it cannot be LI.
4. Be able to prove that any p-dimensional vector space is isomorphic to R^p or C^p, depending on whether it is real or complex.
5. Be able to change coordinates - i.e., given coordinates of a vector relative to one basis, be able to find coordinates relative to another. Be able to construct change-of-basis matrices.
Bases for spaces associated with a matrix - § 5.5
1. Be able to find bases for the row and column spaces of a matrix.
2. Know how to find the rank of a matrix. Be able to show that it is the same as the row rank and the column rank, which are dimensions of the row and column spaces, respectively.
Norms and inner products - § 5.6, 5.7
1. Know the definition of a norm and of a real or complex inner product.
2. Three important norms on R^p or C^p - || v ||₁, || v ||₂, || v ||_infinity . Given a vector, be able to find these norms.
3. Be able to prove the inequalities showing that these are equivalent norms.
4. Be able to define convergence of a sequence using these norms.
5. Be able to prove both Schwarz's inequality and the triangle inequality for real inner product spaces. Be able to find the angle between vectors.
6. Orthogonal and orthonormal sets of vectors. Be able to show that an orthogonal set of nonzero vectors is LI.
Orthogonal projections and Gram-Schmidt - § 5.8.
1. Know the definition of an orthogonal projection onto a subspace V₀. Be able to show that if v' is the orthogonal projection of v onto V₀, then we have
  1. v - v' is orthogonal to V₀.
  2. v' is independent of the particular orthogonal basis used to compute it.
  3. The projection is a linear function of v.
  4. Best approximation properties.
2. Gram-Schmidt procedure.