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
5
32 30 5 8 2 4
5 4
3 16 15 24 24 1 0
8 6 2
Check:
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 trivial solution
c1 0,., c p 0 . The set is termed linearl
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 and Ax 0
Theorem 2
The null space of an m n matrix A is
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 mathematical
systems. Vector spaces in general are a set of o
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 formula is unwieldy for n 3 .
For any n n matrix A and a
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 interchanged in a matrix A to produce a matrix B,
then d
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 by ad bc
c d
For n 2 the determinant of an n n matrix A
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 either all true or all false).
a.) A is an invertible matri
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 matrix A1
with the property that A A1 A1 A I n
If such an i
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
usual protocol for naming in spreadsheets)
a11
a
A 21
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 column of I n is labeled ei .
1 0 0
Example I 3 e1 e 2 e3
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
3 6 2 0
1
1 2 0
So solving Ax b amounts to findi
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 explore the sets of
values that solutions may have. The set
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 of
xn
the column vectors using xi as weights.
Ax x1 v1
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-dimensional vector:
v v1 , v2 , v3 ,., vn Note we can use the not
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) Interchange two rows.
3.) (Scaling) Multiply all entries in a
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.
Theorem 7: The Uniqueness Representation Theorem
Let b1 , b 2
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 of vectors S x1 , x 2 ,., x n , x n 1 , each vector is
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
A
B 1 4 3 C 2 1
4 5 2
5
E
3
2 0 1 4 0 2
2 A 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: Note that by Theorem 10:
Let T : n m be a linear tran
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:
2 0 1 2
0 2 0 2
T u Au
3 1 0 3 2 0 6 6
0 2
2 0 a
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 independent so long as for any weights c1 , c2 , c3
5
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 0 2 5
2 x1 7 x2 x3 0 2 7
4 x 2 x 7 x 0 4 2
2
3
1
2 5
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 2 a a 2
Ax
1 2
0 1 7
7
The product is not defi
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.) Given the same vectors u and v . Display the vectors
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.
Study Guide:
3.) Row reduce the matrix to reduced echelon
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 augmented matrix, following the systemic elimination proc
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 augmented
matrix is not a pivot column, i.e., if and only
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?
0 0 1
Solution:
1 1
u1 u 2 1 1 0 and it is clear