# Dr. Mark V. Sapir

## Exercises

Homework due Class 18.

1. Find infinitely many non-proportional functions f(x) in C[0,1] which form angle Pi/3 with the function x2.
2. Given the 3-vector A=(1, 2, 3), find two non-zero 3-vectors B and C such that B is orthogonal to A and C, and C is orthogonal to A.
"Proof" homework due Class 20
1. Consider the set R2 of all 2-vectors. Define a dot product in R2 in the following way:

Prove that this dot product satisfies the properties of dot products in Euclidean vector spaces and thus R2 with this dot product is a Euclidean vector space.

2. Let V be a Euclidean vector space. A parallelogram is a quadruple of vectors of the form (A,B,A,B). Define the diagonals of a parallelogram and prove that the sum of squares of the lengths of diagonals is equal to the sum of squares of the lengths of the sides of the parallelogram.
3. How would you define the sphere of radius r with center A in an arbitrary vector space? Find all constant functions which belong to the sphere of radius 1 with the center x2 in C[0,1].

### Bonus problems

1. (14 points) Prove or disprove that every linear operator in R2 is a composition (product) of a symmetry, a rotation and a dilation.
2. (7 points) Let us define a dot product on R2 by the following formula:

where p,q,r,s are numbers. Prove that R2 with this dot product is a Euclidean space if and only if r=s, p> 0, q> 0, and r2-pq< 0.

## Exercises

Homework due Class 19.

1. Find the standard matrix of a linear operator R4--> R4 defined by the formula

T(a,b,c,d)=(4a-3b+4c-d, 3a-5b+d, a+d, b+d)

2. Is there a solution of the equation T(v)=(1,1,1,1) where T is the operator defined in Problem 1 (use Maple)?
3. Find the standard matrix for the reflection of R2 about the axis with direction (1,2).

## Exercises

Proof homework due Class 23

1. Suppose that f is the function from R3 to R3 which rotates every vector A counterclockwise about some axis L.

a) Show that f is a linear transformation.

b) Show that the column-vectors in the standard matrix A of f are pairwise orthogonal and their lengths are equal to 1.

2. Find the standard matrix of the linear transformation of R3 which reflects every vector V about the plane
ax+by+cz=0

## Exercises

Computational homework due Class 22
1. Find the standard matrix of a linear transformation from R3 to R2 which takes (1,2,3) to (1,2), (2,0,1) to (2,0) and (1,2,4) to (1,1).
2. Find the standard matrix of a linear operator in R2 which is the composition of the rotation counterclockwise through angle Pi/3, the reflection about the axes y=5x and the projection on the line y=-2x.