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seidi.yamauti
Group Title
[linear algebra] <(x1,x2),(y1,y2)> = x1y1 + tx2y2
For which valor of t, it is an in internal (scalar) product?
 one year ago
 one year ago
seidi.yamauti Group Title
[linear algebra] <(x1,x2),(y1,y2)> = x1y1 + tx2y2 For which valor of t, it is an in internal (scalar) product?
 one year ago
 one year ago

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Traxter Group TitleBest ResponseYou've already chosen the best response.0
The internal product (i.e. dot product) of \[\left(\begin{matrix}x1 \\ y1\end{matrix}\right)\] with \[\left(\begin{matrix}x2 \\ y2\end{matrix}\right)\] is \[\left(\begin{matrix}x1+x2 \\ y1+y2\end{matrix}\right)\] So the only value of t which gives this is t=1
 one year ago

helder_edwin Group TitleBest ResponseYou've already chosen the best response.0
u have to check one by one the properties of an inner product: (assuming an vector space over \(\mathbb{R}\)) \[ \large \langle a,b\rangle=\langle b,a\rangle \] \[ \large \langle a,b+c\rangle=\langle a,b\rangle+\langle a,c\rangle \] \[ \large \langle\alpha a,b\rangle=\langle a,\alpha b\rangle=\alpha\langle a,b\rangle \] \[ \large \langle a,a\rangle\geq0\quad\text{and}\quad \langle a,a\rangle=0\Leftrightarrow a=0 \]
 one year ago

Traxter Group TitleBest ResponseYou've already chosen the best response.0
Ok so I guess inner product isn't just another name you're using for dot product then?
 one year ago

helder_edwin Group TitleBest ResponseYou've already chosen the best response.0
that is not true. u can write: \[ \large \langle a,b\rangle=(a_1\quad a_2)\begin{pmatrix} 1 & 0\\ 0 & t \end{pmatrix} \binom{b_1}{b_2} \]
 one year ago

helder_edwin Group TitleBest ResponseYou've already chosen the best response.0
\[ \large =a_1b_1+ta_2b_2 \]
 one year ago

seidi.yamauti Group TitleBest ResponseYou've already chosen the best response.0
The determinant must be > 0, right? Wich would give the answer t>0. I didn't learn all this procces (I don't really know why, but my teacher didn't explicitly show this transformation of scalar product into matrices product). How I solve by the properties?
 one year ago

helder_edwin Group TitleBest ResponseYou've already chosen the best response.0
not the determinant!! for example u might have the matrix \[ \large \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \] has determinant >0 but the resulting expresion is NOT an inner product.
 one year ago

seidi.yamauti Group TitleBest ResponseYou've already chosen the best response.0
I see. I'm gonna study more of Linear Algebra, for I don't want to ask you to teach me everything about it hahaha. Thank you very much ;)
 one year ago

helder_edwin Group TitleBest ResponseYou've already chosen the best response.0
i think it is a terrific idea. i recomend any of Strang's linear algebra books. there are just great. also hoffman & kunze's
 one year ago

seidi.yamauti Group TitleBest ResponseYou've already chosen the best response.0
This Strang is from Gilbert Strang?
 one year ago

helder_edwin Group TitleBest ResponseYou've already chosen the best response.0
yes. the one and only
 one year ago

seidi.yamauti Group TitleBest ResponseYou've already chosen the best response.0
Understood! Thanks again.
 one year ago
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