Jul 18

Euclidean space geometry: questions for repetition

Euclidean space geometry: questions for repetition

How to study definitions and statements

For a definition, choose the shortest and most geometrical formulation. If it's still difficult to remember, repeat it aloud several times. Speaking aloud sends more energy to your brain. Think out equivalent definitions and consequences. As you do this, new neurons will be created in your brain, and that requires time.

Unlike definitions, statements require proofs. If you use a particular fact in your proof, state it explicitly. My criterion of understanding is this: how could I guess a result like this? Can I generalize it or find an analogy? Repeat the theory in large chunks. Explain clearly, as if lecturing. This will allow you to concentrate on logic, as opposed to memorizing.

Roll up your sleeves

Prove or solve the next statements and exercises.

Statement 1. A scalar product is a function of two vector arguments f(x,y)=x\cdot y such that

1) it is linear in the first argument when the second argument is fixed and linear in the second argument when the first one is fixed (this property is called bilinearity);

2) it is symmetric,

3) the scalar product of a vector by itself is non-negative: x\cdot x\geq0.

4) x\cdot x=0 if and only if x=0.

Statement 2. Once we have a scalar product, we can define an associated norm by \Vert x\Vert =\sqrt{x\cdot x}. It has the properties:

1) \Vert x\Vert \geq 0 for all x,

2) \Vert x\Vert =0 if and only if x=0,

3) the norm is homogeneous of degree 1,

4) Cauchy-Schwarz inequality: |x\cdot y|\leq\Vert x\Vert\Vert y\Vert,

5) triangle inequality: \Vert x+y\Vert \leq\Vert x\Vert +\Vert y\Vert .

Statement 3. Once we have a norm, we can define a distance between two points by \text{dist}(x,y)=\Vert x-y\Vert . It satisfies the triangle inequality \text{dist}(x,y)\leq\text{dist}(x,z) +\ \text{dist}(z,y) for any three points x,y,z.

Statement 4. With a scalar product and the associated norm at hand we can define the notions of orthogonality and cosine of an angle between two vectors. The Pythagoras theorem becomes a simple consequence of definitions.

Exercise 1. Derive the expression for the norm of a linear combination.

Exercise 2. What condition on the centers and radii is sufficient for inclusion of balls B(c,r)\subset B(C,R)?

Exercise 3. Prove that the upper part \{(x_1,x_2)\in R^2: x_2>0\} of the plane is an open set.

Exercise 4. Describe the geometry related to the quadratic equation ax^2+bx+c=0:

1) How do we know if the branches of the parabola y=ax^2+bx+c look upward or downward?

2) What is the link between the discriminant and the following situations: a) the parabola does not touch or cross the x axis, b) the parabola touches (is tangent to) the x axis and c) the parabola crosses the x axis at two different points?

Exercise 5. Prove and interpret the parallelogram law:

\|x+y\|^2+\|x-y\|^2=2\| x\|^2+2\| y\|^2.