E2.3 Express T_b^b as the product of three matrices

In summary, the conversation discusses a linear transformation and how it affects basis vectors in $\alpha$. The resulting vectors will be the columns of the matrix representing $T$ in this basis, which can be found by taking the product of three vectors. The conversation also mentions the lack of responses on differential equations questions.
  • #1
karush
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https://www.physicsforums.com/attachments/8962
ok this is my overleaf homework page but did not do (c) and (d)
this class is over but trying to do some stuff I missed.
I am only auditing so I may sit in again next year...;)
also if you see typos much grateful

I don't see a lot of replies on these DE questions so maybe there isn't an army of eager help?
 
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  • #2
Look at what linear transformation $T\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}x- y \\ y- z\\ 2x+ 3y- 3z\end{bmatrix}$ does to each basis vector in $\alpha$:
$T\begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}= \begin{bmatrix}1- 0 \\ 0- 0\\ 2+ 0- 0\end{bmatrix}= \begin{bmatrix}1 \\ 0\\ 2\end{bmatrix}$.
$T\begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix}= \begin{bmatrix}0- 1 \\ 1- 0\\ 0+ 3- 0\end{bmatrix}= \begin{bmatrix}-1 \\ 1\\ 3\end{bmatrix}$.
$T\begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}= \begin{bmatrix}0- 0 \\ 0- 1\\ 0+ 0- 3\end{bmatrix}= \begin{bmatrix}0 \\ -1\\ -3\end{bmatrix}$.

Those vectors will be the columns of the matrix representing T in this basis.
 
  • #3
So $[T]_\beta^\beta$ would be the product of
$$\begin{bmatrix}1 \\ 0\\ 2\end{bmatrix}
\cdot \begin{bmatrix}-1 \\ 1\\ 3\end{bmatrix}
\cdot \begin{bmatrix}0 \\ -1\\ -3\end{bmatrix}$$
?
 

FAQ: E2.3 Express T_b^b as the product of three matrices

What is the purpose of expressing T_b^b as the product of three matrices?

Expressing T_b^b as the product of three matrices allows for a more efficient and organized way of representing the transformation. It also makes it easier to perform operations and calculations on the transformation.

What are the three matrices involved in expressing T_b^b as a product?

The three matrices involved are the translation matrix, rotation matrix, and scale matrix. These matrices represent the translation, rotation, and scaling components of the transformation, respectively.

How do you determine the order of multiplication for the three matrices?

The order of multiplication for the three matrices is determined by the order in which the transformations are applied. Generally, the order is translation, rotation, and then scaling. This can also be remembered as "TRS", or "Translate, Rotate, Scale".

Can the order of multiplication for the three matrices be changed?

Yes, the order of multiplication for the three matrices can be changed. However, this will result in a different transformation and may not achieve the desired result. It is important to carefully consider the order of multiplication to ensure the correct transformation is achieved.

Are there any limitations to expressing T_b^b as a product of three matrices?

One limitation is that this method can only be used for transformations that involve translation, rotation, and scaling. Other types of transformations, such as shearing or reflection, may require a different method of representation. Additionally, the order of multiplication must be carefully considered to ensure the correct transformation is achieved.

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