A common theme in mathematics is (or seems to be) looking at sets with a particular structure, and then looking at functions between those sets which preserve that structure. In groups we have homomorphisms; in topological spaces we have continuous maps; in general categories we have morphisms. In the particular case of vector spaces, though, there are two particular “structures” we want to preserve: vector addition and scalar multiplication. The maps which preserve these are what we refer to as linear transformations.

Specifically, suppose and are vector spaces over the field . A function is called a linear transformation if for all scalars and for all vectors we have

Note that because of this linearity, a linear transformation is completely determined by how it maps the basis vectors of the domain. Suppose that is a basis for V. Let be any vector in with . We then have

.

So if we know each , we can figure out where any other vector will be sent by . This does not mean that is necessarily a basis for the range, . It could be that , in which case these vectors are linearly dependent and can’t both be in the basis. We do have that span , however, so as long as they’re linearly independent they’ll form a basis.

The main thing we want to notice about linear transformations for right now is that if both and are finite dimensional, then a linear transformation can be represented as a matrix. Suppose that is -dimensional with the basis mentioned above, and that is -dimensional with basis . Note that the properties of matrix multiplication tell us that any matrix defines a linear transformation from to :

Now suppose is any other linear transformation. Suppose that the coordinate vector of with respect to the basis is

Now let . We then have

Thus a linear transformation between finite dimensional vector spaces can be represented as a matrix. Notice that the entries of our matrix depend on our particular chosen bases: if one basis were altered, the matrix would change, even though the transformation is the same. We will denote the matrix representing with respect to the and bases as

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