Math 311-502 - Test I Review

General Information

Test I (Thursday, 9/30/04) will have 6 to 8 questions, some with multiple parts. It will cover chapters 1 and 2 (except for Cramer's rule), and section 3.1A. Please bring an 8½×11 bluebook. Problems will be similar to ones done in class, for homework, or on quizzes. 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

Coordinate vectors - Rⁿ

Linear combination of a set of vectors, standard basis for Rⁿ, span of a set of vectors.
Dot product, norm or length of a vector, angle between vectors, unit vectors, projection onto a vector, components along and perpendicular to a vector, coordinates along a vector.

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 parallelograms and volumes of parallelepipeds using cross and dot products.

Systems & matrices

Linear systems

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.
Solving systems via row reduction.
- Augmented matrix form. Convert a system to and from augmented matrix form.
- Row operations and equivalent systems. There are three types of elementary row operations that give equivalent systems: elementary modification, elementary multiplication, and interchange. Be able to use these to put a matrix in reduced row echelon form. Be able to identify the leading entry in a row, the lead variables and free (non leading) variables.
- Solving systems via row reduction. 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.

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.
Homogeneous systems. Know the connection with solutions to a general system and the corresponding homogeneous system.
Linearly independent (LI) and linearly dependent (LD) 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 find the parametric form of a k-plane; know how it relates to solutions of a linear system of equations.

Inverse of a matrix

Let A be an n×n matrix. Be able to be find the inverse of a matrix or show that a matrix is singular via row reducing [A|I]. Be able to solve Ax = b by inverting A. The conditions below are all equivalent to a matrix having an inverse. Similar conditions hold for A to be singular.

A is invertible.
A is row equivalent to I.
The columns of A are linearly independent.
Ax = 0 has only x = 0 as a solution.
det A ≠ 0.

Determinants

Cofactor expansions. Know the terms minor and cofactor, and how to calculate the determinant of a matrix via its cofactor expansion about any given row or column. Here are immediate consequences.

If A has a row or column of 0's, then det A = 0.
det A^T = det A.
If A is upper triangular, lower triangular, or diagonal, then
det A = a₁₁ a₂₂ ... a_nn.
Alternation. If two rows (or columns) are interchanged in a matrix, the determinant changes sign.
- If A has two identical rows or columns, then det A = 0.
Linear properties. The determinant is a linear function of a row (or column), provided the other rows (or columns) are held fixed. (See below for the definition of linear function.)

Row operations and det(A). Below, A is the original matrix and B is the result of the row operation. Be able to use these to calculate a determinant.

Elementary modification: det B = det A.
Elementary multiplication: If R^A_j = c R^B_j, then det A = c det B.
Interchange: (Same as alternation.) det B = - det A.

Inverses. Be able to determine whether an n×n matrix A is invertible or singular from det A.

Product rule. det(AB)=det(A)det(B). Be able to calculate the determinant of a product using this rule.

Linear functions on Rⁿ

Definition and representation. Be able to show that a given function on Rⁿ is a linear function. Be able to find matrix representations similar to ones we've done in class by means of the representation theorem (pg. 106).