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208 CHAPTER 7 Matrices and Linear Systems

After pivoting about each leading entry, resulting in a matrix K, we need only make all the
leading entries 1 to obtain a reduced row echelon form. If K has leading entry α in row i, multiply
this row by 1/α by entering

F := mulrow(K,i,1/α);
After this is done for all the rows containing a leading entry, the reduced row echelon form
of A results.

SECTION 7.3 PROBLEMS

In each of Problems 1 through 12, find the reduced form of 2 2
6. A =
A and produce a matrix such that A = A R . 1 1
⎛ ⎞
−1 4 6
⎛ ⎞
1 −1 3 7. A = ⎝ 2 3 −5 ⎠
1. A = ⎝ 0 1 2 ⎠ 7 1 1
0 0 0
−3 4 4
8. A =
0 0 0
3 1 1 4
2. A =
0 1 0 0 −1 2 3 1
9. A =
1 0 0 0
⎛ ⎞
−1 4 1 1
⎛ ⎞
0 0 0 0 8 2 1 0
⎜ ⎟
3. A = ⎜ ⎟
⎝ 0 0 0 0 ⎠ 10. A = ⎝ 0 1 1 3 ⎠
0 0 0 1 4 0 0 −3
4 1 −7
⎛ ⎞

1 0 1 1 −1
4. A = 11. A = ⎝ 2 2 0 ⎠
0 1 0 0 2
0 1 0
⎛ ⎞
0 1 ⎛ 0 ⎞
0 0
⎜ ⎟ ⎜ −3 ⎟
5. A = ⎜ ⎟
⎝ 1 12. A = ⎜ ⎟
3 ⎠
⎝ 1 ⎠
0 1 1
7.4 Row and Column Spaces
In this section, we will develop three numbers associated with matrices. These numbers play a
significant role in applications such as the solution of systems of linear equations.
Let A be an n × m matrix of real numbers. Each of the n rows is a vector in R . The span
m
of these row vectors (the set of all linear combinations of these vectors) is a subspace of
m
m
R called the row space of A. This may or may not be all of R , depending on A.The
dimension of the row space of A is the row rank of A.
n
Similarly, the m columns are vectors in R . The span of these column vectors is the
n
column space of A, and is a subspace of R . The dimension of this column space is the
column rank of A.

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October 14, 2010 14:23 THM/NEIL Page-208 27410_07_ch07_p187-246