## seidi.yamauti 3 years ago [linear algebra] <(x1,x2),(y1,y2)> = x1y1 + tx2y2 For which valor of t, it is an in internal (scalar) product?

1. Traxter

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

2. helder_edwin

u have to check one by one the properties of an inner product: (assuming an vector space over $$\mathbb{R}$$) $\large \langle a,b\rangle=\langle b,a\rangle$ $\large \langle a,b+c\rangle=\langle a,b\rangle+\langle a,c\rangle$ $\large \langle\alpha a,b\rangle=\langle a,\alpha b\rangle=\alpha\langle a,b\rangle$ $\large \langle a,a\rangle\geq0\quad\text{and}\quad \langle a,a\rangle=0\Leftrightarrow a=0$

3. Traxter

Ok so I guess inner product isn't just another name you're using for dot product then?

4. helder_edwin

that is not true. u can write: $\large \langle a,b\rangle=(a_1\quad a_2)\begin{pmatrix} 1 & 0\\ 0 & t \end{pmatrix} \binom{b_1}{b_2}$

5. helder_edwin

$\large =a_1b_1+ta_2b_2$

6. seidi.yamauti

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?

7. helder_edwin

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.

8. seidi.yamauti

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 ;)

9. helder_edwin

i think it is a terrific idea. i recomend any of Strang's linear algebra books. there are just great. also hoffman & kunze's

10. seidi.yamauti

This Strang is from Gilbert Strang?

11. helder_edwin

yes. the one and only

12. seidi.yamauti

Understood! Thanks again.