# Thread: Transformation Matrix, finding out B S,T and B T,S

1. ## Transformation Matrix, finding out B S,T and B T,S

Hi guys, started a new topic in university.

We are currently doing different stuff with matrix calculation, but in this case I dont know how to combine each matrix with one another.

Assignment:

I have the bases and

of a 3-dimensional ℝ- vector space V, with:

I need to find out and .

What am I supposed to do? I know that it has something to do with the change of the basis and I understand that you can get the same result if you approach this from two different ways, "similar to homomorphism".

What am I supposed to calculate with and in which way?

2. ## Re: Transformation Matrix, finding out B S,T and B T,S

Do you understand your own notation? It pretty much gives you the answer to half of the problem. The other half is its inverse.

3. ## Re: Transformation Matrix, finding out B S,T and B T,S

I am searching for the bases that go from my vectors S to vectors T I guess. I have trouble understanding which inverse matrix I have to calculate though.

4. ## Re: Transformation Matrix, finding out B S,T and B T,S

First, what does this problem have to do with discrete mathematics? Second, what is $\displaystyle B_{S,T}$? This is not a universally accepted notation. Is this the change of basis matrix from S to T? I've seen, in fact, two different conventions with regards to this. In one, the change of basis matrix from S to T converts coordinates of a vector in T into coordinates of the same vector in S. In this case, the columns of the matrix are coordinates in S of basis vectors from T. So $\displaystyle B_{S,T}$ consists of the column vectors you are given, and it converts coordinates in T into coordinates in S. In the other convention, $\displaystyle B_{S,T}$ converts coordinates in S into those in T. Then you need to write coordinates in T of the basis vectors of S as columns. The resulting matrix is the inverse of the matrix from the first convention. The rule of thumb is that if you want the change of basis matrix to convert coordinates from the first basis to the second one, you need to populate the matrix with coordinates with respect to that second basis, which makes sense.

5. ## Re: Transformation Matrix, finding out B S,T and B T,S

$\displaystyle (t_1)_S = \begin{pmatrix} 1 \\ 2 \\ 0\end{pmatrix}$ means $\displaystyle t_1 = s_1 + 2s_2 + 0s_3$, right? So, what does the matrix $\displaystyle B_{S,T}$ take, and what should it give? Think of it as a linear transformation. What about $\displaystyle B_{T,S}$? What should its input and output be? Describe it in words, then give it some input and find its expected output. For instance, the change of basis from $\displaystyle T$ to $\displaystyle S$ should take $\displaystyle t_1$ to $\displaystyle s_1 + 2s_2 + 0s_3$. Next, solve for $\displaystyle s_1$ in terms of $\displaystyle t_1,t_2,t_3$ (similarly for $\displaystyle s_2,s_3$). That will tell you what the change of basis from $\displaystyle S$ to $\displaystyle T$ should give when you give it $\displaystyle s_1, s_2,$ or $\displaystyle s_3$. Those matrices will be inverse matrices.

6. ## Re: Transformation Matrix, finding out B S,T and B T,S

Thank you very much for your answers guys. I will report tomorrow when I come back home.

7. ## Re: Transformation Matrix, finding out B S,T and B T,S

I finally got it. It took my time to understand, but
B S,T are actually just my vectors (t1, t2, t3) and B T,S was the inverse.