Proving the fourth property of inner products

[Onezimo Cardoso]

Homework Statement

Homework Equations

The Attempt at a Solution

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I could show that the first of the three properties are valid for any value of a,b,c but I couldn’t find a way to show the forth one.

Follow all the procedures I already did:

 

[Ray Vickson]

Homework Statement

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Homework Equations

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The Attempt at a Solution

[/B]
I could show that the first of the three properties are valid for any value of a,b,c but I couldn’t find a way to show the forth one.

Hint: if ${\bf x} = (x_1,0)$, it is easy to show that $\langle {\bf x,x} \rangle > 0$ for any $x_1 \neq 0$. Also: for $x_2 \neq 0$ we have $(x_1,x_2) = x_2(x_1/x_2,1)$, so $\langle {\bf x,x} \rangle = x_2^2 \langle {\bf u,u} \rangle$, where ${\bf u} = (x_1/x_2,1) \equiv (t,1)$.

 

[ehild]

Homework Statement

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Your determinant Δ is wrong, There is x₂² in the first term.

 

[StoneTemplePython]

are you allowed to use spectral theory here?

Your matrix

$\begin{bmatrix} a & b\\ b & c \end{bmatrix}$

is real symmetric. If the determinant is positive (one of your conditions in the iff) and the trace is positive (implied by $a \gt 0$... why?) then it this tells you...

 

[Onezimo Cardoso]

 

[ehild]

The Attempt at a Solution

[/B]
I could show that the first of the three properties are valid for any value of a,b,c but I couldn’t find a way to show the forth one.

Follow all the procedures I already did:

If you factorize Δ you get Δ=4x₂²(b²-4ac)

 

[Ray Vickson]

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Why do you suppose I wrote $\langle {\bf x,x} \rangle = x_2^2 \langle {\bf u,u} \rangle$, where ${\bf u} = (t,1)?$ The fact that $t = x_1/x_2$ does not really matter at all if all you want to know is the sign of $\langle {\bf u,u} \rangle$---think about it!