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 be equally weighted.
1. Find the rank of the matrix
2
1
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 1
1
A=
and e1 = 0
2 2
1 1 b11 1
2 2 b = 0
21
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 vk
for some scalars a1 , a2 , . . . , ak , is called a
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 here can be implemented to solve a linear system
a11 x1
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
in a column to the right of the leading entry of the row
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 linear equations, invert
matrices, compute determinants,
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 W be vector spaces. A transformation T from V to W , wr
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 A = span 0
0
0
(b) By looking at rref(A) we know t
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 ith component of Ax is dened
to be the ith row of A do
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 the following homogeneous system of equations. Express
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 4x1 + hx2 = 8 (a) is empty, (b) contains a unique soluti
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 product is the identity
matrixwhich does nothing to a vecto
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 Elimination within matrices Apply Gaussian Elimination
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 a mathematical structure that has many properties simi
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 weirdness.
Background
You already seen examples of diff
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 Gaussian Elimination to find a matrix inverse
Background
A
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.
Background
By now Im sure you know that matrices are not number
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 = 0 x1 3 + x2 5 + x3 9 = 0 (1) 9 3 0 59 3 x3 0 5 A set
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 presentation and should not be regarded as a substitute for tho