Constuct a 3 x 3 matrix, not in echelon form, whose columns span R^3. Show that the matrix you construct has the desired property.
The columns of a 3 x 3 matrix will span `R^3` if the columns are linearly independent. Hence, in order to construct a matrix that will not span this space, at least two of the columns must be linearly dependent - i.e. rank of the matrix must be less than 3.
For instance, if you choose the first column to be [1; 2; 0] and another column [2; 4; 0], the other column can be anything since these two are already linearly dependent.
Then, there exists a 3-dimensional vector that cannot be expressed as a linear combination of these three vectors.
| 1 0 2 |
| 0 1 0 |
| 2 0 4 |
The column vector (1, 0, 0) cannot be expressed as a linear combination of the columns of these matrix. Hence, it's a 3 x 3 matrix not in row echelon form that does not span `R^3` .