Linear Algebra - Linear Independence Matrices

We can check for the linear independence of a matrix by extending out the matrices into column vectors. For example, checking the linear independence of the following matrices:

|1 1|
|1 1|

|1 2|
|3 4|

|2 3|
|5 7|

|1 4|
|6 8|

We would arrange the entries in the following way. In this way we would know if we could get the entries linearly combine any of the entries to form another one of the entries. This would only work if this row reduced to the identity matrix, if it does not then it is linearly dependent.

|1 1 2 1|
|1 2 3 4|
|1 3 5 6|
|1 4 7 8|

|1 1 2 1|
|0 1 1 3|
|0 2 2 5|
|0 3 5 7|

|1 1 2 1 |
|0 1 1 3 |
|0 0 0 -1|
|0 0 2 -2|

|1 1 2 1 |
|0 1 1 3 |
|0 0 2 -2|
|0 0 0 -1|

|1 0 0 0|
|0 1 0 0|
|0 0 1 0|
|0 0 0 1|

Thus the columns of this matrix are linearly independent, so the matrices are linearly independent. Its not possible to multiply the matrices by a factor and add them to one another in order to get the leftover matrix.