Quiz 1 MA 3330, Type A , spring 2000

1.

Find the distance between P₀=(1,2,3,4) and P₁=( 6,7,8,0).

2.

Find the hyperplane in 4-space which goes through P₀=(1,2,3,4) and perpendicular to v=( 6,7,8,9).

3.

Find the parametric equation of the line passing through (1,2,3,4) and (-1,1,2,-2). Write the parametric equation of the line segment between (1,2,3,4) and (-1,1,2,-2).

4.

True or False
Rⁿ is a field.
R is a field.
Rⁿ has a norm.
R is countable
Rⁿ is countable
Z is countable
Finite sets are countable
Every bounded set in R has glb.
Every bounded set in R has the maximum

5.

Let $\u = < 0,1,-1>, \v = <2,1,1>$. Find the angle between $\u$ and $\v$

6.

Define the angle between $u \in R^n$ and $v \in R^n$ using the norm and scalar product in Rⁿ.

7.

Let , T be the open interval (0 , 1 ) in R.
Answer the following questions for S and T respectively.
Determine if S , T are bounded by giving an upper and lower bound.
Find the maximum element of S ( T resp. ) if it has one.
Find the lub of S (T resp.) if it has one.

8.

Describe the triangular inequality in Rⁿ.

9.

Describe (define) the open ball B( (1,2,3,4), 1) in R⁴.

10.

Describe the convex set intuitivley and give an example of convex set and an example of non convex set.

11.

Find the first 4 terms of the sequence defined by a_n= (-2)ⁿ⁺¹ + (-1)ⁿ.
Describe a_n as a function whose domain is .

12.

Sketch the level curve of f(x,y)=x² + y² at the level 9.

13.

Part II- Bring this part ob Wed.

(a)

Show that an open rectangle whose 4 vertices are (0,0), (1,0), (1,1) and (0,1) is a convex set.

(b)

Show that the set of all functions defined on an open interval (0,1) forms a vector space over R.