Ma 309 Matrix Algebra
Midterm Test #2
Prof. Sawyer Washington Univ. March 28, 2007
Calculators cannot be used. Closed textbook and notes.
Six problems on two pages.
Dierent parts of problems may not b
Lecture 8: Properties of Inverse Matrices
Not all matrices have inverse matrices! The following 2x2 matrix does not have
an inverse because one cannot find the first column of the inverse Ab1 = e1.
1
3.4
Linear Dependence and Span
P. Danziger
1
Linear Combination
Denition 1 Given a set of vectors cfw_v1 , v2 , . . . , vk in a vector space V , any vector of the form
v = a1 v1 + a2 v2 + . . . + ak
Harvey Mudd College Math Tutorial:
Solving Systems of Linear Equations; Row
Reduction
Systems of linear equations arise in all sorts of applications in many dierent elds of study.
The method reviewed
Section 1.2: Row Reduction and Echelon Forms
Echelon form (or row echelon form):
1. All nonzero rows are above any rows of all zeros.
2. Each leading entry (i.e. left most nonzero entry) of a row is
i
1-27-2010
Row Reduction
Row reduction (or Gauss-Jordan elimination) is the process of using row operations to reduce a
matrix to row reduced echelon form. This procedure is used to solve systems of li
Math 309 7
Day 25
Linear Transformations
Linear Transformations are the verbs of linear algebra; they are ways of moving and
twisting vector spaces that preserve linear subspaces.
Denition. Let V and
Solutions to Linear Algebra Practice Midterm
Summer 2011
1. (a) Let x2 = t, then x1 = 2t. Since x3 = x4 = x5 = 0, all solutions have the form
x = (2t, t, 0, 0, 0) = t(2, 1, 0, 0, 0), so
2
1
ker
1
Matrix-vector multiplication, matrix-vector equations
Let A be a matrix and x.
Here is how we dene matrix-vector multiplication: Ax is a vector whose
length is equal to the number of rows of A. The
MA 242 LINEAR ALGEBRA C1, Solutions to First Midterm Exam
Prof. Nikola Popovic, October 5, 2006, 09:30am - 10:50am
Problem 1 (15 points). Determine h and k such that the solution set of x1 + 3x2 = k 4
81
2.5. Inverse Matrices
2.5 Inverse Matrices
Suppose A is a square matrix. We look for an inverse matrix A 1 of the same size, such
that A 1 times A equals I . Whatever A does, A 1 undoes. Their prod
Matrices
Objectives: At the end of this lesson, you should be able to:
1. 2. 3. 4. Create a matrix from a linear system Identify the various kinds of matrices we can create from them. Define Gaussian
Matrix Operations
Objectives: At the end of this lesson, you should be able to:
1. Define the matrix operations. 2. Apply the matrix operations. 3. Create a matrix equation.
Background
Matrices create
Matrix Properties
Objectives: At the end of this lesson, you should be able to:
1. 2. 3. 4. List the matrix properties. Know when they cannot apply! Understand the matrix identity. Be aware of matrix
Matrix Inverse
Objectives: At the end of this lesson, you should be able to:
1. Understand why we want to find a matrix inverse. 2. Understand why we cannot always find a matrix inverse. 3. Apply Gaus
Matrix Determinants
Objectives: At the end of this lesson, you should be able to:
1. Create a matrix determinant from a system or matrix. 2. Calculate the determinant for 22 and 33 matrices1.
Backgrou
Lecture 6: Linear Independence, Spanning, Basis and Dimension. The homogeneous system Ax = 0 can be studied from a dierent perspective by writing them as vector equations; 3 0 2 1 2 3 x1 0 1 3 5 9 x2
These notes closely follow the presentation of the material given in David C. Lay's textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presenta
FIRST EXAM-A Instructions: Begin each of the eight numbered problems on a new page in your answer book. Show your work, and mention theorems when appropriate. 1. [15 Pts] Find the general solution of