As Wiggin42 states, in order for a 3x3 matrix to span `RR^3`, all three of its columns must be linearly independent of one another. By 'spans `RR^3`' it is meant that the 3 columns of the matrix can be linearly combined to produce all possible coordinates (x,y,z) in `RR^3`.
The matrix Wiggin42 gives is an example of one that does not span `RR^3`as its third column is a linear combination of the first two columns (column 3 = column 1 + column 2).``
Another way to show that a 3x3 matrix does not span `RR^3` is to show that its rank is less than 3, which would be the required rank in order for it to span `RR^3` . This can be demonstrated by showing that the matrix cannot be reduced to echelon form. That is, upon row reduction, two or more of the rows in the row-reduced matrix are identical before echelon form is reached (meaning that echelon form cannot in fact be reached)
[Note, all of this applies to any nxn matrix - in order that the matrix spans `RR^n` all n of the columns need to be linearly independent of one another and the rank needs to be the maximum possible, ie n].
To show that Wiggin42's example matrix does not span `RR^3`,row reduce it with the aim of reaching echelon form. The echelon form of a matrix is the result of linear row transformations (row reduction) where the leading diagonal is the unit vector (1,1,1) and all entries to the left of the leading diagonal are zeroes.
The first leading diagonal element is already 1, so we don't need to scale row 1 at all.
Now subtract row 1 from row 2 so that the first element in row 2 is 0. Do the same for row 3:
`([1,2,3],[1,4,5],[1,6,7])` `rightarrow` `([1,2,3],[0,2,2],[0,4,4])`
Now rescale rows 2 and 3 so that the leftmost element is 1:
After these two steps we can see already that rows 2 and 3 are identical, so that the rank of the matrix is 2 and not the full rank, 3. Therefore the example matrix does not span `RR^3`.
In order for a 3 x 3 matrix to span R^3, all three columns have to be linearly independent. Therefore, a 3 x 3 matrix with linearly dependent columns will not span R^3. ie:
[1 2 3
1 4 5
1 6 7]
The third column is a linear combination of the first two columns so there is no way this matrix will span R^3.