# Thread: Question involving linear operators, matrices, and compositions

1. ## Question involving linear operators, matrices, and compositions

Upcoming midterms are giving me very little time to work on assignments such as this one, so I need help with this.

Suppose $\displaystyle V$ is a vector space over a field $\displaystyle F$.

Suppose $\displaystyle B=\{v_1,...,v_n\}$ is a basis for $\displaystyle V$, $\displaystyle T:V\rightarrow V$ is a linear operator with matrix $\displaystyle A\in F^{n\times n}$ with respect to $\displaystyle B$, $\displaystyle U:V\rightarrow V$ is a linear operator with matrix $\displaystyle D\in F^{n\times n}$ with respect to $\displaystyle B$.

a) Prove that the composition $\displaystyle TU:V\rightarrow V$ is a linear. (typo?)

b) Prove that the matrix of $\displaystyle TU$ with respect to $\displaystyle B$ is $\displaystyle AD\in F^{n\times n}$ by proving that the $\displaystyle ji$th entry of the matrix of $\displaystyle TU$ is $\displaystyle \sum_{k=1}^n a_{jk}d_{ki}$.
A hint provided is as follows: Expand $\displaystyle U(v_i)$ then use this expansion to expand $\displaystyle T(U(v_i))$.

This is probably quite easy, but as I said before, I can't devote too much time to this or I'll not be able to properly prepare for my midterms.

2. Originally Posted by Runty
Upcoming midterms are giving me very little time to work on assignments such as this one, so I need help with this.

Suppose $\displaystyle V$ is a vector space over a field $\displaystyle F$.

Suppose $\displaystyle B=\{v_1,...,v_n\}$ is a basis for $\displaystyle V$, $\displaystyle T:V\rightarrow V$ is a linear operator with matrix $\displaystyle A\in F^{n\times n}$ with respect to $\displaystyle B$, $\displaystyle U:V\rightarrow V$ is a linear operator with matrix $\displaystyle D\in F^{n\times n}$ with respect to $\displaystyle B$.

a) Prove that the composition $\displaystyle TU:V\rightarrow V$ is a linear. (typo?)

b) Prove that the matrix of $\displaystyle TU$ with respect to $\displaystyle B$ is $\displaystyle AD\in F^{n\times n}$ by proving that the $\displaystyle ji$th entry of the matrix of $\displaystyle TU$ is $\displaystyle \sum_{k=1}^n a_{jk}d_{ki}$.
A hint provided is as follows: Expand $\displaystyle U(v_i)$ then use this expansion to expand $\displaystyle T(U(v_i))$.

This is probably quite easy, but as I said before, I can't devote too much time to this or I'll not be able to properly prepare for my midterms.

I'm sorry to say that of all the "explanations" I've heard/read, yours is one of the poorest ones ever: if you "don't have time" to

devote to solve a problem then either leave it for later or leave it for good!

Mathematics is a tough subject and it demmands TIME and dedication, no pretexts allowed. If you're in a

rush for your midterms AND also you MUST know this problem's solution then either stop

sleeping/eating/bathing and DEVOTE the problem the time it needs, or else give it up and concentrate on something else.

Never, ever tell a mathematician you need help with a mathematical problem because "you have no time for it"...ever!

Tonio

Pd. Indeed, the problem is a rather basic and VERY important one. You really should sit on it.

3. Originally Posted by tonio
I'm sorry to say that of all the "explanations" I've heard/read, yours is one of the poorest ones ever: if you "don't have time" to

devote to solve a problem then either leave it for later or leave it for good!

Mathematics is a tough subject and it demmands TIME and dedication, no pretexts allowed. If you're in a

rush for your midterms AND also you MUST know this problem's solution then either stop

sleeping/eating/bathing and DEVOTE the problem the time it needs, or else give it up and concentrate on something else.

Never, ever tell a mathematician you need help with a mathematical problem because "you have no time for it"...ever!

Tonio

Pd. Indeed, the problem is a rather basic and VERY important one. You really should sit on it.
Personally, I think I entered the wrong courses.

I have the first part done, but I can't do the second half. I just can't understand ANY of this.

4. Still need help on the second half, as I think I took an incorrect note from my Prof. while seeing him. He moved too fast through the process (accommodating too many people at the same time), so I probably took something down backwards.

Here is what I have so far, though I probably made a mistake somewhere:

First, I expand $\displaystyle U(v_i)=\sum_{k=1}^n d_{ki}v_k=d_{1i}v_1+d_{2i}v_2+...+d_{ni}v_n$

Then, I expand again with the following: $\displaystyle T(U(v_i)=T\left(\sum_{k=1}^n d_{ki}v_k\right)=\sum_{k=1}^n d_{ki}T(v_k)=d_{1i}T(v_1)+d_{2i}T(v_2)+...+d_{ni}T (v_n)$
$\displaystyle =a_{j1}d_{1i}+a_{j2}d_{2i}+...+a_{jn}d_{ni}$
$\displaystyle =\sum_{k=1}^n a_{jk}d_{ki}$

I can tell already that I've done something wrong, such as not explaining my steps or such. But this is all I could come up with given limited material on the subject. If any help could be provided soon, I'd welcome it.