Download Math 200 Spring 2010 March 12 Definition. An n by n matrix E is

Math 200 Spring 2010
March 12
Definition. An n by n matrix E is called an elementary matrix if it is just
one elementary row operation away from In . 


1 0 0
1 0 0
Examples. The matrices E1 =  0 4 0 , E2 =  −2 1 0 , and
0 0 1
0 0 1


0 1 0

E3 = 1 0 0  are elementary matrices.
0 0 1

a b c

Effect of multiplication by an elementary matrix. For M = d e f
g h i



a b c
a
b
c



notice that E1 M = 4d 4e 4f , E2 M = −2a + d −2b + e −2c + f
g h i
g
h
i

d e f
and E3 M =  a b c .
g h i
In general, if an elementary matrix E is obtained from In by a particular
row operation, then multiplying any n-row matrix M on the left by E has
the same effect as performing that row operation directly on M .
Definitions. The span of a set of vectors is the set of all linear combinations of them. The row space of a matrix is the span of its row vectors,
and the column space is the span of its column vectors. So the column
space of a matrix is the same as its image (why?).
Notation. We use the notation rref (A) to signify the reduced row
echelon form of the matrix
A.




1 2 −2 10
1 0 0 2
For example, If A =  3 7 −7 34 , then rref (A) =  0 1 −1 4 .
1 4 −4 18
0 0 0 0
What we can tell from rref (A): Let A be an m by n matrix.
• The number of nonzero rows in rref (A) is the dimension of the row
space of A. Those rows form a basis for the row space of A, which is
a subspace of Rn .
• The number of nonzero rows in rref (A) is also the dimension of the
column space of A, which is a subspace of Rm . To find a basis for the
column space of A, find which columns in rref (A) have leading ones,
and then choose the corresponding column vectors from the original
matrix A.

,

,
So for the matrix A above, the row
space of A is a 2-dimensional subspace
4
of R , with basis { 1 0 0 2 , 0 1 −1 4 }. The
 column
 space of
1
2 3
A is a 2-dimensional subspace of R , with basis  3  ,  7  .
1
4
HOMEWORK DUE MONDAY, MARCH 15:
1. For each of the elementary matrices in the first examples above, find
the inverse. Is the inverse an elementary matrix? If so, what is its
defining elementary row operation? How does that relate to the
defining row operation of the original matrix?
2. For each matrix A, find (i) a basis for ker(A), (ii) a basis for the
image of A, and (iii) a basis for the column space of A.
1 −2
a) A =
−2 4


2 4 −1
b) A =  5 0 3 
1 12 −6


1 0 1 4
c) A =  4 1 1 3 
3 2 0 4
3. If A is an m by n matrix, and rref (A) has k leading ones (meaning
k nonzero rows), what is the dimension of
a) ker(A)?
b) the image of A?
c) the column space of A?
4. Suppose that M is an n by n matrix that is row-equivalent to In .
a) Is M invertible? Explain.
b) What is ker(M )?
c) What is the image of M ?
d) Does M~x = ~b have a solution for every ~b ∈ Rn ?
e) What is the row space of M ?
5. Suppose that C is a 3 by 5 matrix. Fill in the blanks:
a) Multiplication on the left by C defines a linear transformation
from R to R .
b) The dimension of the kernel of C is at least
and at most
.
c) The dimension of the image of C is at least
and at most
.