# Find $\left[T\right]_\beta^\beta$ for NMH{823}

• MHB
• karush
In summary: I start to feel lost.In summary, the author is trying to figure out how to find the transition matrix for a given system. They mention that sometimes it is helpful to have steps shown, but in the end are glad there is a forum to turn to when lost.
karush
Gold Member
MHB
nmh{823}

ok just want to see if I went the right direction with this before I launch into the rref
of which I would use a online calculator to do if I could find one.
$\left[T\right]_\beta^\beta$ should be on the right after rref

Last edited:
Given that $$T\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}x+ 2y- z \\ 2x- y+ z \\ x+ z\end{bmatrix}$$

Then $$T\begin{bmatrix}1 \\ 0 \\ 1 \end{bmatrix}= \begin{bmatrix}1+ 2(0)- 1 \\ 2(1)- 0+ 1 \\ 1+ 1\end{bmatrix}= \begin{bmatrix} 0 \\ 3\\ 2 \end{bmatrix}$$. That is very different from your "$$\begin{bmatrix}2 \\ 0 \\ 2 \end{bmatrix}$$". Am I misunderstanding something?

To write that in basis $$\beta$$ you want to find numbers, A, B, C such that
$$A\begin{bmatrix}1 \\ 0 \\ 1 \end{bmatrix}+ B\begin{bmatrix}1 \\ 2 \\ 1 \end{bmatrix}+ C\begin{bmatrix}1 \\ 1 \\ 0 \end{bmatrix}$$$$= \begin{bmatrix}A+ B+ C \\ 2B+ C \\A+ B \end{bmatrix}$$$$= \begin{bmatrix} 0 \\ 3 \\ 2 \end{bmatrix}$$.

So we have the three equations, A+ B+ C= 0, 2B+ C= 3, and A+ B= 2. From the second equation C= 3- 2B, and from the third, A= 2- B. Putting those into the first equation, A+ B+ C= 2- B+ B+ 3- 2B= 5- 2B= 0. 2B= 5 so B= 2/5. Then C= 3- 4/5= 11/5 and A= 2- 2/5= 8/5. The first column of the transition matrix is $$\begin{bmatrix}\frac{8}{5} \\ \frac{2}{5} \\ \frac{11}{5}\end{bmatrix}$$.

https://www.physicsforums.com/attachments/8893

how did they get these (red)

karush said:
how did they get these (red)
They applied the operator T to the vector using either the explicit form of the matrix T just above or by using the original definition.

$$\displaystyle T \left [ \begin{matrix} 1 \\ 1 \\ 2 \end{matrix} \right ] = \left ( \begin{matrix} 5 & 0 & 1 \\ 3 & 2 & -3 \\ 5 & 0 & 0 \end{matrix} \right ) \left [ \begin{matrix} 1 \\ 1 \\ 2 \end{matrix} \right ] = \left [ \begin{matrix} 5(1) + 0(1) + 1(2) \\ 3(1) + 2(1) - 3(2) \\ 5(1) + 0(1) + 0(2) \end{matrix} \right ] = \left [ \begin{matrix} 7 \\ -1 \\ 5 \end{matrix} \right ]$$

-Dan

Looks obvious now
But i couldn't see that

Sometimes I wish they would show more steps rather that assume things but then ..;)

Just glad there is a good forum to call on

## 1. What does "Find [T]ββ for NMH{823}" mean?

"Find [T]ββ for NMH{823}" is asking for the transformation matrix [T] from the standard basis β to the basis β for the vector space NMH{823}. This matrix represents how the coordinates of a vector in the standard basis would change when expressed in the basis β.

## 2. How do I find the transformation matrix [T]ββ?

To find the transformation matrix [T]ββ, you will need to know the basis vectors for the standard basis β and the basis β for NMH{823}. Then, you can use the formula [T]ββ = [I]ββ^-1, where [I]ββ is the identity matrix with the basis vectors for β as its columns, and ^-1 represents the inverse operation.

## 3. What is the purpose of finding [T]ββ?

The purpose of finding [T]ββ is to be able to convert coordinates of vectors between different bases. This is useful in various applications, such as solving systems of linear equations, performing transformations in geometry, and working with different coordinate systems in physics and engineering.

## 4. Can [T]ββ be different for different vector spaces?

Yes, the transformation matrix [T]ββ can be different for different vector spaces. This is because the basis vectors for each vector space can be different, and the transformation matrix is dependent on the basis vectors.

## 5. Are there any shortcuts or tricks for finding [T]ββ?

There are some shortcuts or tricks that can be used to find [T]ββ, such as using row reduction techniques or using the change of basis formula. However, it is important to understand the concept and formula for [T]ββ in order to use these shortcuts effectively.

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