For a linear transformation ** T**:

(T) = [x |
a_{11}x_{1} + a_{12}x_{2} + ... + a_{1n}x_{n} |
] | |||||

a_{21}x_{1} + a_{22}x_{2} + ... + a_{2n}x_{n} |
|||||||

: | |||||||

a_{m1}x_{1} + a_{m2}x_{2} + ... + a_{mn}x_{n} |
|||||||

= | x_{1}[ |
a_{11} |
] + x_{2}[ |
a_{12} |
] + ... + x[_{n} |
a_{1n} |
] |

a_{21} |
a_{22} |
a_{2n} |
|||||

: | : | : | |||||

a_{m1} |
a_{m2} |
a_{mn} |
|||||

= |
x_{1}a_{1} + x_{2}a_{2}
+ ... + x_{n}a_{n} |

Thus `range T` is given by all the linear combinations
(i.e., for all choices of the coefficients

The column space of a matrix ** A** is

`col A` = {all the linear combinations of columns of

= {

= {

We remark that the column space of an `m` by `n` matrix is a subspace of **R**^{m}.

Example Because

(`a` + `b`, 3`a` - `b`, 2`a` + `b`) = `a`(1, 3, 2) + `b`(1, -1, 1),

the subspace

{(`a` + `b`, 3`a` - `b`, 2`a` + `b`): `a`, `b` ∈ **R**} ⊂ **R**^{3}

given in an earlier example is in fact the columns space of the matrix

= [A |
1 | 1 | ]. |

3 | -1 | ||

2 | 1 |

Notice that the linear transformation in the earlier example is also given by the matrix.

We also note that since the system ** Ax** =

[previous topic] [part 1] [part 2] [next topic]