How can the parallelogram law be proven?

In summary, The conversation discusses various mathematical proofs and properties involving matrices and vectors. The first two questions demonstrate the dot product and its relation to vector lengths, while the third question introduces the parallelogram law. The fourth question involves proving the Vandermonde determinant using cofactors.
  • #1
franz32
133
0
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.
 
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  • #2
[tex](\vec{u}+\vec{v})^2=\vec{u}^2+\vec{v}^2+2\vec{u}.\vec{v}[/tex]

[tex](\vec{u}-\vec{v})^2=\vec{u}^2+\vec{v}^2-2\vec{u}.\vec{v}[/tex]

[tex](\vec{u}+\vec{v})^2 - (\vec{u}-\vec{v})^2 = 4\vec{u}.\vec{v}[/tex]

[tex]\frac{(\vec{u}+\vec{v})^2 - (\vec{u}-\vec{v})^2}{4} = \vec{u}.\vec{v}[/tex]
 
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  • #3
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
 
  • #4
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.
 

1. What is the parallelogram law?

The parallelogram law states that the sum of the squares of the lengths of the four sides of a parallelogram is equal to the sum of the squares of the lengths of the two diagonals.

2. How is the parallelogram law used in mathematics?

The parallelogram law is used in mathematics to prove the equality of two vectors. It also helps in understanding the relationship between the sides and diagonals of a parallelogram.

3. What is the proof of the parallelogram law?

The proof of the parallelogram law involves using the Pythagorean theorem and basic algebra to show that the sum of the squares of the lengths of the four sides is equal to the sum of the squares of the lengths of the two diagonals.

4. Can the parallelogram law be applied to any quadrilateral?

No, the parallelogram law is specific to parallelograms only. It cannot be applied to any other type of quadrilateral, such as a square, rectangle, or trapezoid.

5. What is the practical application of the parallelogram law?

The parallelogram law has practical applications in vector algebra, physics, and engineering. It is used to calculate the resultant of two vectors and to determine the forces acting on an object in equilibrium.

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