MATH 337-102 Lecture #2

1. MATH 337 Lecture #2

2. Slide 1.3- 2 © 2012 Pearson Education, Inc.
PARALLELOGRAM RULE FOR ADDITION
• If u and v in R2 are represented as points in the plane,
then u + v corresponds to the fourth vertex of the
parallelogram whose other vertices are u, 0, and v.

3. Slide 1.3- 3© 2012 Pearson Education, Inc.
ALGEBRAIC PROPERTIES OF Rn
• The vector whose entries are all zero is called the zero
vector and is denoted by 0.
• For all u, v, w in Rn and all scalars c and d:
(i)
(ii)
(iii)
(iv) ,
where denotes
(v)
(vi)
(vii)
(viii)
u v v u  
(u v) w u (v w)    
u 0 0 u u   
u ( u) u u 0     
u ( 1)u
(u v) u vc c c  
( )u u uc d c d  
( u)=(cd)(u)c d
1u u

4. Slide 1.3- 4 © 2012 Pearson Education, Inc.
LINEAR COMBINATIONS
• Example 2: Let , and .
Determine whether b can be generated (or written) as
a linear combination of a1 and a2. That is, determine
whether weights x1 and x2 exist such that
(1)
If vector equation (1) has a solution, find it.
1
1
a 2
5
 
  
 
  
2
2
a 5
6
 
 
 
  
7
b 4
3
 
 
 
  
1 1 2 2
a a bx x 

5. Slide 1.3- 5 © 2012 Pearson Education, Inc.
LINEAR COMBINATIONS
• A vector equation
has the same solution set as the linear system whose
augmented matrix is
. (5)
• In particular, b can be generated by a linear
combination of a1, …, an iff ∃ a solution to the linear
system corresponding to the matrix (5).
1 1 2 2
a a ... a bn n
x x x   

6. Slide 1.3- 6 © 2012 Pearson Education, Inc.
Spans
• Definition: If v1, …, vp are in Rn, then the set of all
linear combinations of v1, …, vp is denoted by Span
{v1, …, vp} and is called the subset of Rn spanned (or
generated) by v1, …, vp. That is, Span {v1, ..., vp} is the
collection of all vectors that can be written in the
form
with c1, …, cp scalars.
1 1 2 2
v v ... vp p
c c c  

7. Slide 1.4- 7 © 2012 Pearson Education, Inc.
Matrix-Vector Multiplication
• Definition: If A is an mxn matrix, with columns a1, …,
an, and if x is in Rn, then the product of A and x,
denoted by Ax, is the linear combination of the
columns of A using the corresponding entries in x as
weights; that is,
.
• Ax is defined only if the number of columns of A
equals the number of entries in x.

8. Slide 1.4- 8 © 2012 Pearson Education, Inc.
MATRIX EQUATION Ax = b
• Theorem 3: If A is an mxn matrix, with columns a1,
…, an, and if b is in Rn, then the matrix equation
has the same solution set as the vector equation
which, in turn, has the same solution set as the
system of linear equations whose augmented matrix
is
.
x bA 
1 1 2 2
a a ... bn n
x x x a   

9. Existence
• For is Ax=b consistent for all b?A =
1 3 4
-4 2 -6
-3 -2 -7
é
ë
ê
ê
ê
ù
û
ú
ú
ú

10. Slide 1.4- 10 © 2012 Pearson Education, Inc.
EXISTENCE OF SOLUTIONS
• The equation has a solution iff b is a linear
combination of the columns of A.
• Theorem 4: Let A be an mxn matrix. Then the
following statements are logically equivalent (i.e.,
either they are all true or they are all false).
a. For each b in Rm, the equation Ax = b has a
solution.
b. Each b in Rm is a linear combination of the
columns of A.
c. The columns of A span Rm.
d. A has a pivot position in every row.
x bA 

11. Matrix-Vector Multiplication
• If the product Ax is defined, then the ith entry
in Ax is the sum of the products of
corresponding entries from row i of A and
from the vector x.

12. Slide 1.4- 12 © 2012 Pearson Education, Inc.
PROPERTIES OF THE MATRIX-VECTOR
PRODUCT Ax
• Theorem 5: If A is an mxn matrix, u and v are vectors
in Rn, and c is a scalar, then
a.
b. .
(u v) u v;A A A  
( u) ( u)A c c A

13. Slide 1.5- 13 © 2012 Pearson Education, Inc.
HOMOGENEOUS LINEAR SYSTEMS
• A system of linear equations is said to be
homogeneous if it can be written in the form Ax=0,
where A is an mxn matrix and 0 is the zero vector in
Rn.
• Such a system Ax=b always has at least one solution,
namely, x = 0 (the zero vector in Rn).
• This zero solution is usually called the trivial solution.
• The homogeneous equation Ax=0 has a nontrivial
solution iff the equation has at least one free
variable.

14. Slide 1.5- 14 © 2012 Pearson Education, Inc.
HOMOGENEOUS LINEAR SYSTEMS
• Example 1: Determine if the following homogeneous
system has a nontrivial solution. Then describe the
solution set.
1 2 3
1 2 3
1 2 3
3 5 4 0
3 2 4 0
6 8 0
x x x
x x x
x x x
  
   
  

15. Slide 1.5- 15 © 2012 Pearson Education, Inc.
HOMOGENEOUS LINEAR SYSTEMS
• Example 2: Determine if the following homogeneous
system has a nontrivial solution. Then describe the
solution set.
10x1 – 3x2 – 2x3 = 0

16. Slide 1.5- 16 © 2012 Pearson Education, Inc.
SOLUTIONS OF NONHOMOGENEOUS
SYSTEMS
• When a nonhomogeneous linear system has many
solutions, the general solution can be written in
parametric vector form as one vector plus an
arbitrary linear combination of vectors that satisfy
the corresponding homogeneous system.
• Example 2: Describe all solutions of Ax=b, where
3 5 4
3 2 4
6 1 8
A
 
   
 
  
7
b 1
4
 
  
 
  

17. Slide 1.5- 17 © 2012 Pearson Education, Inc.
SOLUTIONS OF NONHOMOGENEOUS
SYSTEMS
• Theorem 6: Suppose the equation Ax=b is consistent
for some given b, and let p be a solution. Then the
solution set of Ax=b is the set of all vectors of the
form w=p+vh, where vh is any solution of the
homogeneous equation Ax=0.

18. Slide 1.5- 18 © 2012 Pearson Education, Inc.
WRITING A SOLUTION SET (OF A
CONSISTENT SYSTEM) IN PARAMETRIC
VECTOR FORM
1. Row reduce the augmented matrix to reduced
echelon form.
2. Express each basic variable in terms of any free
variables appearing in an equation.
3. Write a typical solution x as a vector whose entries
depend on the free variables, if any.
4. Decompose x into a linear combination of vectors
(with numeric entries) using the free variables as
parameters.