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January 8th, 2013, 03:51 AM   #1
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Linear Transformations and Matrices

In the page that I attached, (for theorem 2.126) I was a bit confused about why $T(x_j)= \sum_p a_{pj}y_{p}$.

Also, for theorem 2.127, I was confused about $F(1_V)= I$. In order to show that something is a ring homomorphism, we need to show that F(1)=1, right? But I'm confused here, because aren't we supposed to say F(I) = I (since the identity matrix of V is I)? Why do they instead say $F(1_V)= I$. Isn't $1_V$ the identity map...then how is it the identity element?

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 January 8th, 2013, 06:28 AM #2 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: Linear Transformations and Matrices Sorry but I see no attachment. The sum you give, $T(x_j)= \sum_p a_{pj}y_p$ makes no sense to me. I think it should be $y_j= T(x)= \sum_p a_{pj}x_j$. For you second question, what distinction are you making between "1" and "I"? They are both commonly used for the multiplicative identity in an algebraic structure, such as ring.
January 8th, 2013, 08:36 AM   #3
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Re: Linear Transformations and Matrices

Quote:
 Originally Posted by HallsofIvy Sorry but I see no attachment. The sum you give, $T(x_j)= \sum_p a_{pj}y_p$ makes no sense to me. I think it should be $y_j= T(x)= \sum_p a_{pj}x_j$. For you second question, what distinction are you making between "1" and "I"? They are both commonly used for the multiplicative identity in an algebraic structure, such as ring.

I'm sorry, I didn't realize that I didn't attach anything...

Also, for $1_V$...I thought that meant the identity map, that's why I was confused. It makes sense now, since you said it was the identity element...but then why don't they just write F(I)=I?

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