Linear Algebra

126 Chapter 2. Vector Spaces
3.6 Definition The column space of a matrix is the span of the set of its
columns. The column rank is the dimension of the column space, the number
of linearly independent columns.
Our interest in column spaces stems from our study of linear systems. An
example is that this system
c1 +3c2 +7c3 = d1
2c1 +3c2 +8c3 = d2
4c1
c2 +2c3 = d3
+4c3 = d4
has a solution if and only if the vector of d’s is a linear combination of the other
column vectors,
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 3 7
⎜
c1
⎜2⎟
⎜
⎟
⎝0⎠
+ c2
⎜3⎟
⎜
⎟
⎝1⎠
+ c3
⎜8
⎟
⎝2⎠
4 0 4
=
⎛ ⎞
d1
⎜d2⎟
⎜ ⎟
⎝d3⎠
meaning that the vector of d’s is in the column space of the matrix of coefficients.
3.7 Example Given this matrix,
⎛
1
⎜
⎜2
⎝0
3
3
1
⎞
7
8 ⎟
2⎠
4 0 4
to get a basis for the column space, temporarily turn the columns into rows and
reduce.
⎛
⎞
1 2 0 4
⎝3
3 1 0⎠
7 8 2 4
−3ρ1+ρ2
⎛
⎞
1 2 0 4
−2ρ2+ρ3
−→ −→ ⎝0
−3 1 −12⎠
−7ρ1+ρ3
0 0 0 0
Now turn the rows back to columns.
⎛ ⎞
1
⎜
〈 ⎜2
⎟
⎝0⎠
4
,
⎛ ⎞
0
⎜ −3 ⎟
⎝ 1 ⎠
−12
〉
The result is a basis for the column space of the given matrix.
3.8 Definition The transpose of a matrix is the result of interchanging the
rows and columns of that matrix. That is, column j of the matrix A is row j of
A trans , and vice versa.
d4