How does change of basis on linear transformation matrix work

Question

Wrestling with conversion of bases on a linear transformation. According to the books, "In order to enable matrix multiplication to give us the result of some linear T applied to a vector v, we must make sure T and v are both expressed in a common basis. In that case, w = MT · v only makes sense if both vectors and all the columns of MT are expanded along that common basis."

I was trying to wrap my arms around this so I thought I'd start with a degenerate case and see how it plays out.

Ok, let's let the basis of a one dimensional space be (1) Let the transformation be the identity transformation, so the matrix of transformation is [1]. Let's apply that to the vector (1):

[1]•(1)=(1)

Now let's see what happens if we change the basis to (2)

So according to the rule above, we view the transformation matrix as a list of column vectors, and translate each of those column vectors the new basis

That yields [1/2] for the transformed matrix.

Then we translate our vector to the new basis, which is (1/2). So the result is [1/2]•(1/2)=(1/4)

Which is not what the book guarantees I'm sure it's something simple but I'm missing it

Answer 1

I see my mistake. You need to (1) first apply the transformation to each of the new basis vectors, then (2) for each resulting basis vector change the basis to the new basis, then (3) augment those into a new transformation matrix. Then (4) apply that to the vector, expressed in the new basis

So in my example,
Step 1. [1]x(2)=(2),
Step 2. change to the basis (2) which yields (1) in the new basis
Step 3. New transformation matrix is [1]
Step 4. The original vector (1) becomes (1/2) in the new basis.
So the transformation in the new basis is
[1]x(1/2)=(1/2) in the new basis, which translates to (1) in the original basis as expected

Answer 2

The matrix of identity is the same in all bases.

To ensure this in the present case ($1$-dimensional), consider the change of basis matrix $P$: its columns are the components of the vectors in the new basis w.r.t. the old basis, i.e. it is the matrix $P=(1/2)$. On the other hand, the formula to have the expression of a linear map in the new basis ($A'$) in function of its matrix in the old basis ($(A)$) is: $$A'=P^{-1}AP.$$

Here, $P=(1/2)$, so $P^ {-1}=(2)$, and $$A'=(2)A(1/2)=(2)(1)(1/2)=(1).$$