MHB What is T_alpha^beta in matrix form using e1 and e2?

[karush]

pic to avoid typos
View attachment 8865
ok my my baby step on this was
a) then
\begin{align*}\displaystyle
\begin{bmatrix}
1\\0
\end{bmatrix}&=e_1\\
\begin{bmatrix}
1\\1
\end{bmatrix}&=e_1+e_2
\end{align*}
so
$$\displaystyle
\left[ T\right]_\alpha^\beta
=\begin{bmatrix}
1&1\\
0&1
\end{bmatrix}$$

 

[Evgeny.Makarov]

my baby step on this was

Baby steps are OK provided they are in the right direction. Could you copy the definition of "the change of basis matrix from $\alpha$ to $\beta$ and $[T]^\alpha_\beta$ from your textbook? Also, what textbook are you using? I am asking because different sources mean different things by the direction of change: "from $\alpha$ to $\beta$". In some books this means the matrix is composed of coordinates in $\alpha$ of vectors in $\beta$, and others say that the matrix convert coordinates of a vector in $\alpha$ into coordinates of the same vector in $\beta$. Unfortunately, these directions are opposite of each other. Some books even say, "the change of basis matrix between $\alpha$ and $\beta$", which is not specific enough, in my opinion. Similarly, there are different conventions on which index goes up and which goes down in $[T]^\alpha_\beta$.

\begin{align*}\displaystyle
\begin{bmatrix}
1\\0
\end{bmatrix}&=e_1\\
\begin{bmatrix}
1\\1
\end{bmatrix}&=e_1+e_2
\end{align*}

It's not good to use undefined notations such as $e_1$ and $e_2$. In every new thread you are supposed to define the notations you are using.

 

[karush]

Baby steps are OK provided they are in the right direction. Could you copy the definition of "the change of basis matrix from $\alpha$ to $\beta$ and $[T]^\alpha_\beta$ from your textbook? Also, what textbook are you using? I am asking because different sources mean different things by the direction of change: "from $\alpha$ to $\beta$". In some books this means the matrix is composed of coordinates in $\alpha$ of vectors in $\beta$, and others say that the matrix convert coordinates of a vector in $\alpha$ into coordinates of the same vector in $\beta$. Unfortunately, these directions are opposite of each other. Some books even say, "the change of basis matrix between $\alpha$ and $\beta$", which is not specific enough, in my opinion. Similarly, there are different conventions on which index goes up and which goes down in $[T]^\alpha_\beta$.

It's not good to use undefined notations such as $e_1$ and $e_2$. In every new thread you are supposed to define the notations you are using.

So what would be your first step?

 

[Evgeny.Makarov]

My first step would be to consult the textbook and verify the definitions of the change of basis matrix and the notation $[T]^\alpha_\beta$. Without this we may make some guesses, but why do work that may need to be redone later?

 

[karush]

ok, I don't buy textbooks and there is no free download for this bookThe question I am asking is supposedly very standard and common.

 

[Evgeny.Makarov]

It is indeed common, but there is a couple of ways to interpret it, and I wanted to avoid extra work. Anyway, what textbook is it?

It's not the first time I see you trying to solve a problem without knowing all needed definitions. This is... not very smart, to put it mildly. Mathematics has enough legitimate difficulties and does not need unwise ways of studying added to them.

 

[karush]

It all new to me
Just trying my best

 

[karush]

It is indeed common, but there is a couple of ways to interpret it, and I wanted to avoid extra work. Anyway, what textbook is it?

It's not the first time I see you trying to solve a problem without knowing all needed definitions. This is... not very smart, to put it mildly. Mathematics has enough legitimate difficulties and does not need unwise ways of studying added to them.

View attachment 8867

 

[Evgeny.Makarov]

The best I could find out is the following. The change of basis matrix from $\alpha$ to $\beta$ consists of coordinates of vectors from $\beta$ written as columns. The coordinates are taken in $\alpha$. This matrix converts the coordinates in $\beta$ of any vector into the coordinates in $\alpha$ of the same vector.

The matrix of an operator $T$ with respect to $\alpha$ and $\beta$, written $[T]^\alpha_\beta$ consists of coordinates of $T(v)$ written as columns for vectors $v$ in $\beta$. The coordinates are taken in $\alpha$. This matrix converts the coordinates in $\beta$ of any vector $v$ into the coordinates in $\alpha$ of $T(v)$. In particular, $^\alpha_\beta$ where $I$ is the identity operator converts coordinates in $\beta$ into coordinates of the same vector in $\alpha$, i.e., it is the change of basis matrix from $\alpha$ to $\beta$.

Now back to the original problems.

a) Find $[T]^\beta_\alpha$. To follow the definition, you should take the basis vectors from $\alpha$, i.e., the standard basis, apply $T$ to them, find the coordinates of the results in $\beta$ and then write them as columns.

b) Find $[T]^\beta_\beta$. Do the same thing, but start with basis vectors from $\beta$ instead of $\alpha$.

c) Find $^\alpha_\beta$. Take basis vectors from $\beta$, find their coordinates in $\alpha$ and write as columns. In fact, you have found this matrix in the rest of post #1.

d) Find $^\beta_\alpha$. Take basis vectors from $\alpha$, find their coordinates in $\beta$ and write as columns.

Off topic: If you don't have a textbook, then you should attend every class and take extra complete notes during lectures even if you don't have a clue what is being discussed. If the professor uses slides, ask to download them. Also, if you have not done so, go to an office hour and ask for an advice. Describe what you understand and what you don't and what the reasons for that are.

 

[karush]

View attachment 8876View attachment 8877
View attachment 8878View attachment 8879
Finally got pdf of textbook
Ok I didn't know how they got b)

 

[Evgeny.Makarov]

What I wrote in the previous post was correct, judging from the textbook.

Ok I didn't know how they got b)

It's not nice to say "I did not understand how you got to your conclusion" to a person who for ten minutes was explaining to you how he got to his conclusion. In this context by such person I mean the textbook author. It's OK, though, to point at a specific phrase or equation and ask, "How did you get this statement from the previous one?"

The solution to $Ax=b$ is $x=A^{-1}b$. So one needs to multiply both sides by $A^{-1}$. This can be done gradually by performing elementary row operations because each operation is equivalent to multiplication from the left by a certain matrix. Therefore when after performing these operations the left-hand side becomes the identity matrix, it means that it has been multiplied by $A^{-1}$. Since the right-hand side was subjected to the same transformations, it was also multiplied by $A^{-1}$ and is therefore equal to $A^{-1}b$. A similar thing can be done when you need to solve several systems of equations with the same matrix but different right-hand sides. Then all right-hand sides $b$ are collected into a single matrix.