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Thread: Question involving linear operators, matrices, and compositions

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by Runty View Post
    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.
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  3. #3
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    Quote Originally Posted by tonio View Post
    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.
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  4. #4
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    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.
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