To look at how linear operators transform vectors in coordinate form, we first modify how we represent vectors - we represent their components as a column matrix. So in 2-space, this representation is

,

for example

.

The usual rules for vector addition and scalar multiplication in component form transfer immediately to the addition and scalar multiplication of vectors in column matrix form.

In 3-space, the columns have three entries, but otherwise, the idea is exactly the same.

Remember that a linear operator on 2-space is determined completely by how it transforms the basis vectors i and j. Suppose we have a linear operator with

.

If we then translate the equation T(xi + yj) = xT(i) + yT(j) into column vector form and do a few matrix calculations, we get

.

Notice what's happened here: we've translated the operation of T on the vector into the multiplication of that vector by a 2x2 matrix whose columns are the images of i and j. You could go backwards here: given any 2x2 matrix

,

define a linear operator on 2-space by defining its effect on i and j:

.

Then multiplying a vector by M is the same as applying the linear operator T to the vector.

This process works equally well for vectors in 3-space. We can formalize the conclusions above as follows:

For any linear operator T on 2-space, let M be the matrix with columns T(i) and T(j). Then M is called the standard matrix of T, and for any vector u in 2-space, T(u) = Mu. Any 2x2 matrix is the standard matrix of the linear operator in 2-space whose columns are the transforms of i and j.

For any linear operator T on 3-space, let M be the matrix with columns T(i), T(j). and T(k). Then M is called the standard matrix of T, and for any vector u in 3-space, T(u) = Mu. Any 3x3 matrix is the standard matrix of the linear operator in 3-space whose columns are the transforms of i, j and k.

In the diagram below, the red vector is the transformation of the blue one, using the linear operator defined by the 2x2 standard matrix. You can change the blue vector by dragging its head, or change the standard matrix by clicking on its entries and typing.

Experiment with the diagram above to see if you can figure out how vectors in 2-space transform if the rows of the standard matrix are proportional to each other.