When we talk about linear transformations, we need to consider how the matrix that represents them can change based on the basis we pick for our vector spaces.

Simply put, a linear transformation is a function that takes vectors from one space to another. The way these vectors are shown can vary depending on the basis used.

Let’s say we have a linear transformation called ( T: V \rightarrow W ), where ( V ) and ( W ) are vector spaces. The matrix representing this transformation changes depending on the bases you choose for both spaces.

For example, if we have the bases for ( V ) as ( {v_1, v_2, \ldots, v_n} ) and for ( W ) as ( {w_1, w_2, \ldots, w_m} ), the matrix for ( T ) with these bases is called ( [T]{B_A,B_B} ). But if you switch to new bases ( {v_1', v_2', \ldots, v_n'} ) for ( V ) and ( {w_1', w_2', \ldots, w_m'} ) for ( W ), the new matrix ( [T]{B_C,B_D} ) can look very different.

To find the matrix representation, you take each basis vector in ( V ) and apply the transformation ( T ) to it. The result is some vector in ( W ), and you show that in terms of the chosen basis. The values you find become the columns of your matrix. This leads us to an important point: the way vectors are represented can create very different matrices for the same transformation.

For example, think about a simple rotation in a two-dimensional space, like ( \mathbb{R}^2 ). When we use the standard basis (which is like the x and y axes), it looks like this:

$$[T]_{Standard} = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}$$

Now, if you switch to a different basis, one that's rotated compared to the standard basis, the matrix representing this rotation will change a lot. Even though the transformation still performs the same rotation, how it looks in the new basis is different. It’s kind of like telling the same story in different languages—the main idea might be the same, but how you express it can vary widely.

The way different bases relate is managed by something called change of basis matrices. If ( P ) is the change of basis matrix, you can find the new matrix representation like this:

$$[T]_{new} = P^{-1}[T]_{old}P$$

So, understanding how different bases affect the matrix representation is really important. It not only changes how you calculate the transformation but also how you interpret it and make calculations easier. This knowledge helps mathematicians and engineers effectively work with transformations in different situations.