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**1. Homework Statement**

Reduce ##xy+zy## to diagonal form.

**2. Homework Equations**

The desired diagonal form is ##Q(\vec{x})=(\alpha_1(\vec{x}))^2+...+(\alpha_k(\vec{x}))^2-(\alpha_{k+1}(\vec{x}))^2-...-(\alpha_{k+l}(\vec{x}))^2,## where ##\alpha_i## are linearly independent linear functions. Also known as 'changing the variables in quadratic form'.

**3. The Attempt at a Solution**

##xy+zy=y^2+yx+yz+(\frac{x}{2})^2+\frac{xz}{2}+(\frac{z}{2})^2-((\frac{x}{2})^2+\frac{xz}{2}+(\frac{z}{2})^2)-y^2=(y+\frac{x}{2}+\frac{z}{2})^2+(\frac{x}{2}+\frac{z}{2})^2-y^2##, but now

##\alpha_1=y+\frac{x}{2}+\frac{z}{2}##

##\alpha_2=\frac{x}{2}+\frac{z}{2}##

##\alpha_3=y##

are linearly dependent. I have tried several different substitutions without success. There's also the matrix method which I am not familiar with. This rather simple problem is giving me headache.

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