Homework Assignment #5 Part Two
Section 2.2 Inverse of Matrix Problems 1,7,9,11,13,19
8 6
1.) Find the inverse of
5 4
1 d b
provided ad bc
ad bc c a
3
8 6
1 4 6 2
Therefore the inverse of
is
Section 4.3 Linearly Independent Sets; Bases.
Definition: A set of vectors v1 , v 2 ,., v p in a vector space V is linearly independent if
the vector equation c1 v1 c2 v 2 . c p v p 0 has only the tri
Section 4.2 Null Spaces, Column Spaces, & Linear Transformations
Definition The null space of an m n matrix A, written NulA is the set of all solutions to
the homogeneous equation Ax 0 .
NulA x : x n
Section 4.1 Vector Spaces and Subspaces
Introduction: The group of order 5 as a mathematical entity (set plus operation).
Vector spaces and subspaces are an extension of ideas from n into other mathem
Section 3.3 Cramers Rule, Volume, and Linear Transformations
Cramers Rule
Cramerr rule is used in many applications. It can be used to study how the solution to
Ax b is affected by changes in b. The f
Section 3.2 Properties of Determinants
Theorem 3
Let A be a square matrix.
1.) If a multiple of one row of A is added to another row of A,
to produce a matrix B, then det A det B.
2.) If two row are i
Section 3.1 Introduction to Determinants
Notation Definition:
Aij is the matrix produced by deleting the ith row and the jth column from matrix A.
a b
The determinant of 2 2 matrix A det
is given b
Section 2.3 Characterizations of Invertible Matrices
Theorem 8 (Invertible Matrix Theorem IMT)
Let A be an n n matrix. Then the following statements are equivalent:
(i.e., for a given A, they are eith
Section 2.2 The Inverse of a Matrix
The inverse of a nonzero real number
a is denoted by a 1. It is defined by the property that a a 1 1
An n n matrix A is said to be invertible if there exists a matr
Section 2.1 Matrix Operations
Recall the two ways to represent an m n matrix.
In term of the entries of A:
Note that the entries are names by their row designation first. (This is the opposite of the
Section 1.9 The Matrix of a Linear Transformation
Definition: The identity matrix, I n , of an n n matrix is a matrix with all ones on the
main left to right diagonal and zeros elsewhere. The ith colu
Section 1.8 Introduction into Linear Transformations
Another way to view the matrix equation Ax b is as an object and an operation
(multiplication).
Examples:
2 4
8
3 6 2 12
3
1 2 4
2 4
0
Section 1.5 Solution Sets of Linear Systems
In previous sections we have seen that linear systems may be inconsistent or the may
have a single solution or an infinite number of solutions. We now explo
Section 1.4 The Matrix Equation Ax = b
Definition: Let A v1
v2
, v n be a m n matrix with n columns equal to the
x1
x
vectors v i and let the vector x 2 n then define Ax as the linear combination o
Section 1.3 Vector Equations
Review of HW #7 Section 1.2
Matrix representation of vectors:
There are several ways to represent a vector.
An ordered n-tuple that lists the components of an n-dimensiona
Section 1.2 Row Reduction and Echelon Forms
Review of elementary row operations of matrices:
1.) (Replacement) Multiply a row by a constant and add the result to another row.
2.) (Interchange) Interch
Section 4.4 Coordinate Systems
This section involves imposing real number coordinate systems on a vector space in order
to render it more usable and to make it more analogous to known entities.
Theore
Section 4.5 The Dimension of a Vector Space
Theorem 9
If a vector space V has a basis b1 , b 2 ,., b n , then any set in V containing more than
n vectors must be linearly dependent.
Proof. Given a set
Homework Assignment #5 Part One
Section 2.1 Matrix Operations Problems 1,7,13,15,19,21,23
1.) Given the following matrices, Find the indicated sums or products if they are defined:
2 0 1
7 5 1
1 2
Homework Assignment #4 Part Two
Section 1.9 The Matrix of a Linear Transforamtion Problems: 1,3,7,13,19,21
3
5
1
2
4
and T e 2 2
1.) T : by T e1
3
0
1
0
Find the standard matrix of T .
Solution
Homework Assignment #4 Part One
Section 1.8 Linear Transformations Problems: 1,5,7,13,19,23
2 0
1.) Let A
, and define T : 2 2 by T x Ax.
0 2
1
a
Find the image under T of u and v
3
b
Solution:
Homework Assignment #3 Part Two
Section 1.7 Linear independence
Problems: 1,5,7,13,15,21,25
1.) Determine if the vectors are linearly independent:
5 7 9
0 , 2 , 4
0 6 8
The vectors are linearly
Homework Assignment #3 Part One
Section 1.5 Solution Sets of Linear Systems
Problems: 1,3,7,11,13,19
1.) Determine if the following system of linear equations has a nontrivial solution.
2 x1 5 x2 8 x3
Homework Assignment #2 Part Two
Section 1.4 Matrix Equation Ax=b
Problems: 1,5,7,13,19
1.) Compute the product using the definition and also using the row-vector rule for
computing Ax .
4 2 3
3
1 6
Homework Assignment #2 Part One
Section1.3 Vector Equations
Problems: 1,3,7,9,13,17,19
1
3
1.) Given u and v Find u v and u 2 v
2
1
1 3 4
1
3 1 6 5
u v and u 2 v 2
2 1 1
2
1 2 2 4
1
3
3
Homework Assignment #1 Part Two
Section1.2 Row Reduction & Echelon Form
Problems: 1,3,7,13,17,19
1.) Determine which matrices are in reduced echelon form and which others are only in
echelon form.
Stu
Homework Assignment #1 Part One
Section 1.1 Systems of Linear Equation
Problems: 1,5,7,13,17,19,23,25
1.) Solve the system of equations by using elementary row operations on the equations or
on the au
Linear Algebra Theorems
Chapter One
Theorem 1
Each matrix is row equivalent to one and only one matrix in RREF.
Theorem 2
1. A linear system is consistent if and only if the rightmost column of the au
Section 6.2 Orthogonal Sets
Definition A set of vectors u1 , u 2 ,., u p is called an orthogonal set if
ui u j 0 for all i j .
1 1 0
Example: Is the set u1 , u 2 , u3 1 , 1 , 0 an orthogonal set?