How can the parallelogram law be proven?

[franz32]

Hello everyone! =)

This might be a good challenge to everyone here... =)

1.) Let A be an n X n matrix and let x and y be vectors in R^n.
Show that Ax .y = x.(A^T)(y), where "." means dot product and T is 'transpose'.

2. show that u.v = (1/4)// u + v //^2 - (1/4)//u - v//^2
where u and v are vectors; "." means dot product and
//...// denote the length of a vector.

3. Prove the parallelogram law: // u + v //^2 + // u - v //^2 =
2 //u//^2 + 2 //v//^2.

4. Prove the Vandermonde determinant.

 

[himanshu121]

$$(\vec{u}+\vec{v})^2=\vec{u}^2+\vec{v}^2+2\vec{u}.\vec{v}$$

$$(\vec{u}-\vec{v})^2=\vec{u}^2+\vec{v}^2-2\vec{u}.\vec{v}$$

$$(\vec{u}+\vec{v})^2 - (\vec{u}-\vec{v})^2 = 4\vec{u}.\vec{v}$$

$$\frac{(\vec{u}+\vec{v})^2 - (\vec{u}-\vec{v})^2}{4} = \vec{u}.\vec{v}$$

 

[franz32]

Hello there!

Hi.

You're right. Well, if you pretty know the property involved on the right side, you could expand the following...

//u + v//^2 = //u//^2 + //v//^2 + 2 (u.v) and
//u - v//^2 = //u//^2 + //v//^2 - 2 (u.v) Thus

u.v = (1/4)//u//^2 + (1/4)//v//^2 + 1/2(u.v) - (1/4)//u//^2 - (1/4)//v//^2 + 1/2(u.v)

u.v = 1/2(u.v) + 1/2(u.v)
= u.v

 

[franz32]

Parallelogram Law

In #3, it is very easy, as long as you know the equivalent of
//u + v//^2 and //u - v//^2 (look on the previous replies.)

the sum of the two yields:

//u//^2 + //v//^2 + 2(u.v) + //u//^2 + //v//^2 - 2(u.v)

and thus = 2//u//^2 + 2//v//^2

In Vandermonde determinant, use the cofactor for easier proving.