# Question involving linear operators, matrices, and compositions

• Oct 12th 2010, 09:55 AM
Runty
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 $V$ is a vector space over a field $F$.

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

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

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

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.
• Oct 12th 2010, 12:18 PM
tonio
Quote:

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 $V$ is a vector space over a field $F$.

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

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

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

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.
• Oct 15th 2010, 11:32 AM
Runty
Quote:

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.
• Oct 17th 2010, 05:13 PM
Runty
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 $U(v_i)=\sum_{k=1}^n d_{ki}v_k=d_{1i}v_1+d_{2i}v_2+...+d_{ni}v_n$

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

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.